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MELANIE KOETSVELD
JOE MARSIGLIO
LYN ELMS
DINA ANTONIOU NSW CURRICULUM
8A Area review
8B Surface area review
8C Volume review
8D Surface area of pyramids and cones ADV STN
8E Volume of pyramids and cones ADV STN
8F Surface area and volume of spheres ADV STN
Diagnostic pre-test
Take the diagnostic pre-test to assess your knowledge of the prerequisite skills listed below.
Interactive skillsheets
After completing the diagnostic pre-test, brush up on your knowledge of the prerequisite skills by using the interactive skillsheets.
✔ Converting between units of length
✔ Using Pythagoras’ theorem
✔ Area of a triangle
✔ Area of quadrilaterals
✔ Area of a circle
• Solves problems involving the surface area of right prisms and practical problems involving the area of composite shapes and solids (MA5-ARE-C-01)
• Solves problems involving the volume of composite solids consisting of right prisms and cylinders (MA5-VOL-C-01)
• Applies knowledge of the surface area of right pyramids and cones, spheres and composite solids to solve problems (MA5-ARE-P-01)
• Applies knowledge of the volume of right pyramids, cones and spheres to solve problems involving related composite solids (MA5-VOL-P-01)
© NESA
✔ Calculator
By the end of this topic you will be able to...
✔ calculate the area of composite shapes
✔ calculate the area of sectors.
• To convert between units of length, multiply or divide by theappropriate conversion factor.
• A unit of area can be converted to another unit of area by multiplying or dividing by the appropriate conversion factor.
• 1 ha (hectare) = 10 000 m2
• To find the area of a composite shape:
1 split the shape into individual parts
2 calculate any missing dimensions
3 calculate the areas of the individual parts
4 add or subtract the areas to find the total area.
Inter-year links
Year 7 8E Area of a triangle
Year 8 8D Area of a circle
Year 9 6A Area of composite shapes
• The area of a sector of a circle is θ 360° × πr 2, where θ is the angle, in degrees, between the two radii forming the sector.
Convert each length or area measurement below to the unit shown in brackets.
a Identify the correct conversion factor. To convert from metres (m) to centimetres (cm), multiply by 100.
b Identify the correct conversion factor. To convert from millimetres (mm) to metres (m), first divide by 10 (to convert to centimetres), then divide by 100 (to convert to metres).
c Identify the correct conversion factor. To convert from square kilometres (km2) to square metres (m2) multiply by 10002
d Identify the correct conversion factor. To convert from square millimetres (mm2) to square centimetres (cm2), divide by 102
a 5.6 m = (5.6 × 100) cm = 560 cm
b 120 mm = (120 ÷ 10 ÷ 100) cm = 0.12 m
c 1.2 km2 = (1.2 × 10002) m2 = 1 200 000 m2
d 320 mm2 = (320 ÷ 102) cm2 = 3.2 cm2
THINK
1 Redraw the shape with dashed lines, splitting it into individual shapes for which you can calculate the area.
2 Label any unlabelled dimensions you know for each individual shape. The total length of the rectangle is 15 cm and the length either side of the triangle is 5 cm. The cut-out triangle must have a base of 5 cm (because 15 – 2 × 5 = 5).
3 Calculate the area of each individual shape.
4 Calculate the required area by subtracting the area of the triangle from the area of the rectangle.
1 Identify the values of θ and r
2 Substitute the values into the formula for the area of a sector.
3 Calculate the area, correct to one decimal place, including the appropriate unit.
✔ There can be more than one way to split a composite shape into individual component shapes. But the way you split up the shape should not change the total area.
✔ When being asked to round to a given number of decimal places, the rounding should only take place at the final stage of your work and not in any intermediate steps. For example, if you are calculating the area of a composite shape, use the full numbers on your calculator when finding the areas of the individual shapes and only round after adding or subtracting.
1–3, 4(a, c, e), 5–7, 8(a, c, e), 9, 10
1, 3(c, e, f), 4(b, d, f), 6, 7(b, d, f), 8, 9, 11, 13, 15
For questions involving π, use the π key on your calculator.
1, 3(c, f), 4(d, f), 6, 7(d, f), 10, 12–16
1 Convert each length or area measurement below to the unit shown in brackets.
a 7 m (cm)
d 29 mm2 (cm2)
g 34 200 000 cm2 (km2)
b 4500 m (km)
e 24 000 m2 (ha)
h 10.22 km (cm)
c 8.4 km2 (m2)
f 0.88 m (mm)
i 3 km2 (ha)
2 Match each shape below with a formula from this list that could be used to find its area:
a
d
3 Calculate the shaded area of each of these composite shapes. Give your answers to one decimal place where necessary.
4 Calculate the shaded area of each of these composite shapes. Give your answers in the units given in the brackets, rounded to one decimal place where necessary.
a (cm2)
(cm2)
c (cm2)
(mm2)
e (cm2)
(cm2)
5 The front view of this house is made of a number of basic shapes.
a Identify each shape.
b Calculate the area of each window pane you identified in part a.
c There are four window panes of equal size. Explain how you can calculate the area of all four window panes if you know the area of only one window pane.
d The owners wish to paint the front of the house (including the door, roof and chimney) and need to find the total area to be painted.
Using your knowledge of how to find the area of a composite shape, explain how to calculate the required area. (Remember that the windows won’t need to be painted.)
e Use your answers for parts c and d to help you find the area of the house to be painted.
f If 1 L of paint covers 15 m2, how many litres of paint do the owners need to purchase? Give your answer to the nearest half litre.
6 Calculate the area of each of these sectors correct to one decimal place.
7 Use your knowledge of Pythagoras’ theorem to find any unknown lengths for each of these composite shapes and then calculate the shaded area of each one. Give your answers to one decimal place where necessary. Hint: Check that all measurements are written in the same units.
8 An annulus is the region between two circles of different sizes which share the same centre.
a Explain how you can find the area of the annulus on the right.
b Find the area of the garden bed correct to one decimal place.
c Find the area of the following annuli, correct to one decimal place.
9 A quadrant is a sector with θ = 90° and a semicircle is a sector with θ = 180°.
a Write a formula for the area of a quadrant with radius r
b Write a formula for the area of a semicircle with radius r
c How do your formulas in parts a and b relate to the formula for the area of a circle with radius r?
d Explain why the area of a sector is θ 360 × πr2, where θ is the angle, in degrees, between the two radii forming the sector.
10 A rectangle has an area of 720 cm2. One of its sides is five times as long as another.
a If x is the length of the smaller side, draw a diagram of the rectangle and label the sides.
b Use this information to find the dimensions of the rectangle.
c If the dimensions are all doubled, what is the area of the new rectangle?
d How does the area of the new rectangle compare to the area of the old rectangle?
e If the dimensions of the original rectangle are all tripled, what is the area of the new rectangle? Compare this area to the original area of 720 cm2.
f Explain what happens to the area of the rectangle when the dimensions are all: i quadrupled
ii multiplied by a factor of n.
11 A clock has a minute hand that is 7 cm long and an hour hand that is 5 cm long.
a What is the difference, correct to one decimal place, between the areas covered by the hands when they each complete a full revolution?
b The time shown on the clock face on the right is 2 o’clock. Calculate the angle of the sector between the minute hand and the hour hand.
c Find the difference in the areas that would be covered by the hands if they each moved through the sector from the 12 to the 2 on the clock face. Give your answer correct to one decimal place.
d Find the difference in the area that would be covered by the hands if they each moved through the smaller sector created between them when the time is 7 o’clock. Give your answer correct to one decimal place.
12 A design based on the ancient yin-yang symbol is shown on the right.
a What would be the relationship between the radius of the complete circle and the radius of the semi-circles if they were formed inside it?
b If the radius of the full circle is 24 cm, find the area shaded blue correct to one decimal place (yin).
c Find the area of the circle’s white section (yang) without performing any further calculations.
13 The area of a parallelogram is A = bh, where b is the base length and h is the perpendicular height. Find the total shaded area of this hash symbol.
14 Consider the target on the right. The radius of the inner circle is 2 cm and the edge of each ring is a further 2 cm from the centre of the inner circle, as shown.
a Find the area, as an exact value, of the shaded region.
b Find the area, as an exact value, of the unshaded region.
c Find the ratio of the shaded area to the unshaded area. 2046810
15 A regular hexagon can be thought of as the composite of six congruent equilateral triangles. Give your answers to the following questions as exact values.
a Use Pythagoras’ theorem to find the height, as an exact value, of an equilateral triangle with side lengths 10 cm.
b Hence, find the area of an equilateral triangle with side lengths 10 cm.
c Hence, find the area of a regular hexagon with side lengths 10 cm.
d Find the area of a regular hexagon with side lengths of a.
16 A trapezium is a quadrilateral with a pair of parallel sides. The area of a trapezium is A = h 2(a + b), as shown below. b
a h
Demonstrate that the trapezium area formula works using the following methods.
a Considering a trapezium as the composite of two triangles, as shown below.
h
a
b Considering a trapezium as the composite of one rectangle and two triangles, as shown below. Hint: you can combine the two triangles into one triangle, since they have the same height. b a
c Considering a trapezium as a truncated triangle, as shown below. Hint: The two triangles are similar and so you can substitute out x in the area equation by considering ratios.
By the end of this topic you will be able to ...
✔ calculate the surface area of prisms and cylinders
✔ calculate the surface area of composite solids consisting of prisms and cylinders.
Inter-year links
Year 7 7F 3D objects
Year 8 8D Area of a circle
Year 9 6B Surface area of prisms
• The surface area (SA) of a 3D object is the total area of its outer surface. This is the sum of the areas of the faces (or surfaces) of that object.
• Prisms are 3D objects that have a uniform cross-section that is a polygon.
➝ All faces of a prism are flat.
➝ Cubes, rectangular prisms and triangular prisms are all common examples of prisms.
• Cylinders are 3D objects with a uniform circular cross-section and a curved face. The surface area of the curved face of a cylinder is equal to the circumference (2πr) of either end multiplied by the height (h):
Area of curved face of a cylinder = 2πrh
• The surface area of a cylinder (the area of the two circular ends and the curved surface) can be found using the formula:
SA = 2πr2 + 2πrh
• A net is a 2D plan that can be folded to form a 3D object. Nets can be used to show all the faces of 3D objects and hence to help calculate the surface areas of those objects.
Net of a rectangular prism
Net of a triangular prism
• A composite solid consists of two or more simple solids.
• To calculate the surface area of a composite solid, identify all faces of the composite solid, add the area of each face to find the surface area of the composite solid.
• The area where two solids join should not be included in the calculation of surface area.
Calculate the surface area of this triangular prism.
THINK
1 Identify the number of faces and the shapes of the faces of the triangular prism.
2 Calculate the area of each face. The triangular faces are isosceles triangles, meaning the two rectangles sharing common sides with them are identical.
WRITE
The prism has five faces: two identical rectangular faces, two identical triangular faces and a rectangular base.
rectangular faces: A = 10 × 15 = 150 cm2 each
rectangular base: A = 4 × 15 = 60 cm2
triangular faces: A = 1 2 × 4 × 9.8 = 19.6 cm2 each
3 Add the areas and include the appropriate unit.
SA = 2 × 150 + 60 + 2 × 19.6 = 399.2 cm2
Calculate the surface area of this cylinder, correct to one decimal place. 10 mm 32 mm
THINK
1 Identify the values of r and h for this cylinder.
2 Substitute the values you identified into the formula for the surface area of a cylinder and simplify.
3 Calculate the result using the π key on your calculator. Write the answer correct to one decimal place and include the appropriate unit.
WRITE
r = 10 mm h = 32 mm
SA = 2πr2 + 2πrh
= 2 × π × 102 + 2 × π × 10 × 32
= 200π + 640π = 840π
= 2638.937...
≈ 2638.9 mm2
Calculate the surface area of this composite object.
1 Identify the number of faces and the shapes of the faces of the solid.
2 Calculate the area of each face.
The solid has nine faces: three pairs of identical rectangular faces (including the pair of sloping faces), a pair of identical triangular faces and the rectangular base.
rectangular faces (front and back): A = 3.4 × 12 = 40.8 m2 each
rectangular faces (sides): A = 3.4 × 5.5 = 18.7 m2 each
rectangular faces (sloping faces): A = 6.6 × 5.5 = 36.3 m2 each
rectangular base: A = 12 × 5.5 = 66 m2
triangular faces: A = 1 2 × (6.1 – 3.4) × 12 = 1 2 × 2.7 × 12 = 16.2 m2 each
3 Add the areas and include the appropriate unit.
SA = 2 × 40.8 + 2 × 18.7 + 2 × 36.3 + 66 + 2 × 16.2 = 81.6 + 37.4 + 72.6 + 66 + 32.4 = 290 m2
✔ If you can’t remember the formula to calculate the surface area of a prism or cylinder, consider drawing a net of the solid and calculating the area of the shapes making up the net.
✔ In some instances, the area where the faces of solids join in a composite solid is the same as the area of the top face of the combined solid. Identifying this can help with your calculations. For example, in the diagram on the right the surface area of the composite solid is the surface area of the bottom cylinder plus just the curved surface area of the top cylinder because its exposed circular end is the same as the joined surface calculated as part of the surface area of the bottom cylinder.
✔ The surface area of a prism can also be found by using the formula:
SA = 2 × area of base + perimeter of base × height
1 Calculate the surface area of each of these prisms.
2 Calculate the surface area of each of these triangular prisms correct to one decimal place. Hint: Use Pythagoras’ theorem.
3 If the base of this hexagonal prism has an area of 41.52 cm2, calculate surface area.
4 Calculate the surface area of each of these cylinders correct to one decimal place.
5 Calculate the surface area of each of these composite solids. Give your answers correct to one decimal place where appropriate.
6 Each die in this stack is the shape of a cube with side lengths of 15 mm.
a Calculate the surface area of one die.
b Use your answer for part a to calculate the total surface area of nine separate dice.
c When stacked, the nine dice form a rectangular prism. Using the information you have been given about each die, draw the rectangular prism in your books and label its dimensions.
d Calculate the surface area of the rectangular prism made of nine dice.
e Find the difference between your answers to parts b and d. Explain why these answers differ. Support your explanation with calculations.
7 The net of a 3D object is not unique. Here are two nets of the same triangular prism.
a Draw the triangular prism corresponding to the two nets above.
b Draw a third net of the triangular prism which is different to the two given above.
c Are all diagrams consisting of two equilateral triangles with 2 cm sides and three 4 × 2 cm rectangles nets of the same triangular prism?
8 An open pipe has a length of 33 cm and a diameter of 55 mm.
a Calculate the surface area of the outer surface of the pipe correct to the nearest cm2
b To find your answer for part a, did you need to use the entire formula for the surface area of a cylinder: SA = 2πrh + 2πr 2? Explain.
9 Find the surface area of the outside of an open cylinder that has a radius of 4.5 cm and a height of 9.9 cm. Give your answer correct to one decimal place.
10 The teachers at a kindergarten want to add two cylindrical poles to the playground. One pole is to be painted yellow and the other red. The vertical poles have the same diameter of 125 mm, but they vary in height.
a The yellow pole is 1.8 m tall. Calculate the area needing to be painted yellow correct to one decimal place. Assume that the top end and the side of the pole needs painting but the end in the ground does not.
b The red pole is double the height of the yellow pole. Compared to the amount of yellow paint needed, do the teachers need to purchase double the amount of red paint in order to paint this second pole? Why or why not?
c Calculate the area of the pole needing to be painted red correct to one decimal place. Does your result support your answer to part b?
11 Find the surface area of the following cut cylinders. Give your answers in cm2 correct to one decimal place.
12 Calculate the surface area of each of these composite solids correct to one decimal place.
13 Calculate the surface area of this prism.
14 Seven cylinders with the same height but different diameters are stacked together as shown on the right. The top cylinder has a diameter of 1 cm and the diameter of each cylinder beneath it increases by 1 cm each time. The bottom cylinder has a diameter of 7 cm. If each cylinder is the same height, 1.5 cm, calculate the surface area when the cylinders are sitting one on top of the other, as shown. Give your answer correct to one decimal place.
15 A company commissions an artist to make a giant version of their logo, an H, for the foyer of their office building. The specifications are in the diagram on the right. The H will be made of wood and then every face of the H will be painted, except the two rectangles on the ground.
a Calculate the surface area of the H, not including the two rectangles touching the ground. Answer in square metres.
b The H will be painted blue, as that is the company’s colour. The company plans to use a premium blue paint which is sold in 2 L cans. Each litre of this paint is enough to paint one coat on 4 square metres. How many cans of paint must they buy if they paint the H with a triple coat?
By the end of this topic you will be able to ...
✔ calculate the volume of prisms and cylinders
✔ calculate the volume of composite solids consisting of prisms and cylinders.
• Volume is measured in cubic units, such as mm3, cm3 and m3.
Inter-year links
Year 7 8F Volume and capacity
Year 8 8E Volume and capacity
Year 9 6D Volume of composite solids
• A unit of volume can be converted to another unit of volume by multiplying or dividing by the appropriate conversion factor.
• The volume of any prism can be found using the formula V = Ah, where A is the area of the base and h is the height of the prism.
➝ The height is perpendicular to the base.
➝ When dealing with a triangular prism, subscripts can be used to represent the two heights, for example h1 and h2, as shown in the diagram of a triangular prism on the right, where h1 is the height of the triangular base.
• The volume of a cylinder can be found by multiplying the area of the base (πr 2) by the height (h).
V = πr 2h
• The volume of a composite solid can be found by calculating the volumes of the individual component solids and adding them together.
• The capacity of a 3D object is a measure of how much the object can hold. A container with an inside volume of:
1 cm3 holds 1 mL of liquid
1000 cm3 holds 1 L of liquid ➝ 1 m3 holds 1 kL of liquid.
Calculate the volume of this prism.
THINK
1 Identify the base of the triangular pr ism and the perpendicular height.
2 Calculate the area of the base.
WRITE
The base is a triangle with base length 9 cm and height 8 cm. The height (h) of the prism is 22 cm.
A = 1 2bh
= 1 2 × 9 × 8
= 36 cm2
3 Substitute the value of the area of the base and the perpendicular height into the formula for the volume of a prism.
4 Include the appropriate unit in your answer.
h = 22 cm
V = bh
= 36 × 22 = 792 cm3
Calculate the volume of this cylinder. Give your answer to one decimal place. 24 cm 15 cm
THINK
a Calculate the radius measurement of the circular base.
b Substitute the values for r and h into the formula for the volume of a cylinder.
c Calculate using the π key on your calculator, round to one decimal place, and include the appropriate unit in your answer.
WRITE
r = D ÷ 2
= 24 ÷ 2
= 12 cm
h = 15 cm
V = πr2h
= π × 122 × 15
= 2160π
= 6785.840...
≈ 6785.8 cm3
Helpful hints
✔ If the dimensions of a 3D object are given using different units, convert all lengths to the required unit specified in the question before you do anything else.
1, 2(c–i), 3, 4, 5(a, c), 6(a, c, e, g), 7, 8, 9(b), 12, 15
1(b, d, f, h, j), 2(d, e, i, j, k, l), 3(c–e), 5(c), 6(b, d, f, h), 9(a, b), 10, 11, 17 1(g–j), 2(i–l), 6(g-i), 9(b, c), 13, 14, 16, 18, 19
1 Convert each of these volume measurements to the unit shown in brackets.
2 Calculate the volume of each of these prisms.
3 Calculate the volume of each of these cylinders. Give your answers correct to one decimal place.
4 Find the capacity in millilitres (mL) or litres (L) of each of the prisms in question 2 correct to one decimal place.
5 Calculate the volume of each of these prisms.
6 Calculate the volume of each of these composite solids. Where appropriate, give your answers correct to one decimal place. Hint: Identify the individual solids first.
7 A round hatbox has a diameter of 23 cm and a height of 12.5 cm. Calculate its volume correct to one decimal place.
8 In this toy house, constructed from different shaped blocks, there are three common solids that can be identified.
a Identify the three solids.
b Explain how you can calculate the surface area of the toy house.
c Find the surface area of the toy house correct to the nearest cm2.
d Find the volume of wood used to make this toy house, correct to the nearest cm3
9 Use Pythagoras’ theorem to help you calculate the volume of each of these prisms. Give your answers correct to one decimal place where appropriate.
10 This piece of cheese is in the shape of a right-angled triangular prism.
a Calculate its volume correct to one decimal place.
b If 1 cm3 of cheese has a mass of 0.001 kg, find the mass of the cheese, in kilograms, correct to four decimal places.
c This particular cheese costs $21 per kilogram. What is the cost of the piece of cheese shown to the nearest cent?
11 A rectangular fish tank is 45 cm long, 20 cm wide and 50 cm tall.
a Calculate the volume of the fish tank.
b What is the capacity of the fish tank if it is filled to the brim? Give your answer in millilitres and litres. (Remember: 1 cm3 = 1 mL)
c The fish tank should only be filled to a level 6 cm from the top. How many litres of water are required to fill the tank to that height?
12 The volume of a particular prism is 2744 cm3. If the prism is a cube, what is its side length?
13 a If a cylinder has a volume of 1649.34 cm3 and a radius of 5 cm, what is the height of the cylinder, to the nearest centimetre?
b If a cylinder has a volume of 207 cm3 and a height of 5.7 cm, find the radius of its circular base, to the nearest millimetre.
14 Jack installs two cylindrical water tanks. The first tank is 2 m tall and the diameter of its base is 1.5 m.
The second tank is 1.5 m tall and the diameter of its base is 2 m.
a Which tank has the larger capacity?
b What is the difference in the capacity of the two tanks? Give your answer to the nearest litre.
a A particular cylinder has a radius of 8 cm and a height of 15 cm. Calculate its volume.
b If the radius of the cylinder from part a is halved but the height remains the same:
i calculate the volume
ii describe how this volume compares to the volume of the original cylinder.
c If the radius of the cylinder from part a is doubled but the height remains the same:
i calculate the volume
ii describe how this volume compares to the volume of the original cylinder.
d If the radius of the cylinder from part a remains the same but the height is halved:
i calculate the volume
ii describe how this volume compares to the volume of the original cylinder.
e If the radius of the cylinder from part a remains the same but the height is doubled:
i calculate the volume
ii describe how this volume compares to the volume of the original cylinder.
16 A swimming pool is the shape of a rectangular prism. A semi-circular spa is built onto one end of the pool, as shown in the diagram on the right.
a How much water is required if the pool and spa are filled to capacity? Give your answer to the nearest litre.
b To make the pool and spa functional, the water levels must be below the top edge. If the pool is filled to 70% of its depth and the spa is filled to 60% of its depth, how much water is required, to the nearest litre?
17 The plastic pipe on the right is hollow.
a What shape is the base of the pipe? Calculate its area correct to one decimal place.
b Use the area of the base to calculate the volume of plastic used for this pipe.
c Can you think of a different method for finding the volume? Explain your answer.
18 Nicko wishes to make an open box from a 30 cm by 30 cm piece of red cardboard. He decides to cut out a square of length 2 cm from each corner and fold up the sides to form the box. The edges of the box will be joined by sticky tape, so no tabs are needed to create the box.
a Draw a diagram showing the piece of cardboard with the corners cut out. Label the dimensions.
b Determine the length, breadth and height of the box once Nicko has folded up the sides.
c Calculate the volume of the box.
d Would the volume of the box increase or decrease if he cut out 3 cm square corners from the piece of cardboard before forming the rectangular prism? Explain your answer.
19 These nuts have the same dimensions but different shapes. One is circular and the other is in the shape of a regular hexagon. They both have a depth of 6.2 mm.
a How much steel is required to manufacture each nut? Give your answers to one decimal place.
b Which nut requires more steel?
Take the checkpoint quiz to check your knowledge of the first part of this chapter.
1 Convert each length and area measurement to the unit shown in brackets.
a 45 mm (cm) b 400 cm (m)
c 4.53 m (mm) d 4000 mm2 (cm2)
e 5.2 km2 (m2) f 68 000 cm2 (m2)
2 Find the area of the following composite shape.
3 Find the area of the following composite shapes. Give your answer correct to the nearest whole number.
4 Use Pythagoras’ theorem to help determine the area of the following shape.
5 Find the surface area of the following cylinder correct to one decimal place.
6 Use Pythagoras’ theorem to help find the surface area of the following prism correct to one decimal place.
7 A cube with side lengths of 10 cm sits on top of a square-based rectangular prism, as shown below. Find the surface area of the composite solid.
8 Calculate the volume of the following cylinder correct to one decimal place.
9 Find the volume of this rectangular prism in:
10 The following prism has a cross-sectional area of 123.11 cm2. Find the volume of the prism.
By the end of this topic you will be able to ...
✔ calculate the surface area of right pyramids and right cones.
• A pyramid is a 3D object with a polygon as its base. All other faces of a pyramid are triangles that meet at the apex (highest point) of the pyramid.
➝ Pyramids are named according to the shape of their base.
➝ A right pyramid has an apex directly above the centre of its base.
➝ An oblique pyramid does not have its apex over the centre of its base.
• The surface area of a pyramid is the sum of the areas of all of its faces.
Year 8 8B Area of triangles and rectangles
Year 9 6C Surface area of cylinders
• A tetrahedron is a pyramid with a triangular base. A regular tetrahedron has four triangular faces that are all congruent equilateral triangles.
• If the length of the base and the slant edge of a pyramid are known, Pythagoras’ theorem can be used to find the height of a triangular face.
• A cone is a 3D object with a circular base and a curved surface that tapers from the circular base to the apex.
• A right cone has its apex directly above the centre of its base.
• The surface area of a right cone with radius r and slant height s is given by the formula: SA = πr 2 + πrs
• The net of a cone consists of a circle and a sector of a circle.
Note: In this topic, only right pyramids and right cones are considered.
Calculate the surface area of the right rectangular pyramid.
THINK
1 Identify the number and shape of the faces of the pyramid. Since it is a right rectangular pyramid, the triangular faces opposite each other are congruent.
2 Calculate the area of each face.
20 cm
WRITE
The pyramid has five faces: one rectangular face and two pairs of congruent triangular faces.
rectangular face: A = 22 × 15 = 330 cm2
front triangular face: A = 1 2 × 22 × 20 = 220 cm2
side triangular face: A = 1 2 × 15 × 18 = 135 cm2
3 Add the areas of the faces together and include the appropriate unit.
SA = 330 + 2 × 220 + 2 × 135 = 1040 cm2
Calculate the surface area of the tetrahedron on the right. The apex of the tetrahedron is directly above the centre of the 9.9 cm edge.
THINK
1 Identify the number and shapes of the faces of the tetrahedron.
WRITE
The base is an isosceles right-angled triangle:
b = 7 cm, h = 7 cm
There are two identical triangular faces:
b = 7 cm, h = 12.5 cm
There is one other triangular face:
b = 9.9 cm, h = 10.2 cm
2 Calculate the area of each face.
base: A = 1 2 × 7 × 7 = 24.5 cm2
triangular face 1: A = 1 2 × 7 × 12.5 = 43.75 cm2 each
triangular face 2: A = 1 2 × 9.9 × 10.2 = 50.49 cm2
3 Add the areas of the faces together and include the appropriate unit.
SA = 24.5 + 2 × 43.75 + 50.49 = 162.49 cm2
Calculate the surface area of this cone correct to one decimal place.
THI n K
1 Determine the radius and the slant height of the cone.
2 Write the formula for the surface area of a cone. Substitute values for r and s and simplify.
3 Calculate the result using the π key on your calculator. Write the answer correct to one decimal place and include the appropriate unit.
Helpful hints
✔ If a tetrahedron is not regular, you’ll need to work out the area of more than one triangular face to calculate the surface area.
✔ If the base of a right pyramid is a regular polygon, then the surface area can also be found using the formula:
SA = base area + 1 2 × perimeter of base × slant height.
✔ The surface area of a right cone with radius r and slant height s is given by the formula:
SA = πr 2 + πrs
1 Calculate the surface area of each of these pyramids.
2 To calculate the surface area of each of these right pyramids, first find the height of a triangular face. Give your answers to one decimal place where necessary.
3 At the main entrance to the Louvre Museum in Paris, France, a pyramid has been constructed with a square base of length 35 m. Each identical sloping triangular face has a height of 27 m and is made of glass segments. Find the surface area of the glass walls of the pyramid.
4 The Red Pyramid in Dahshur, Egypt, was originally covered in white limestone. It has a slant edge of length 187 m and a square base of side length 220 m.
a Draw a diagram of this pyramid and label its dimensions.
b Find the surface area of the pyramid that was covered with limestone. Hint: Do not include the base in your calculations.
5 Calculate the surface area of each of these tetrahedrons correct to one decimal place. Hint: Use Pythagoras’ theorem to find any missing heights.
6 Calculate the surface area of each of these cones correct to one decimal place.
7 A square-based right pyramid has a surface area of 561 cm2. If the length of the base is 11 cm, find the height of all the triangular faces.
8 A cone has a radius of 15.5 cm and a height of 9.5 cm. Use Pythagoras’ theorem to find the slant height of the cone correct to one decimal place and its surface area.
9 a What is the surface area of a regular tetrahedron with a side length of 4 cm?
b Develop a formula to determine the surface area of a regular tetrahedron with a side length of x cm.
10 A party hat is in the shape of a cone. The diameter of the base is 12 cm and the slant height is 18 cm. Find how much cardboard is needed to make this party hat assuming there is no overlap. Give your answer correct to one decimal place. Hint: Do you need to use all parts of the formula for the total surface area of a cone? An eight-sided die is in the shape of an octahedron.
a How many equilateral triangles form the octahedron?
b If the side length of the octahedron is 14 mm, find the total surface area of the octahedron correct to one decimal place.
c The octahedron resembles two identical square pyramids sitting base to base. Calculate the surface area of one pyramid and double your answer.
d Are your answers for parts b and c equal? Why or why not?
12 Calculate the surface area of each of these composite solids. Give your answers correct to one decimal place.
13 The tip of a hollow cone is sliced through and removed. The remaining object is called a truncated cone. If the removed tip was one-third of the height of the whole cone, find the outer surface area of the truncated cone correct to one decimal place. (Assume the cone includes a base.)
14 An open cone can be made from a circle sector by curving the shape to join the two straight edges. The centre of the circle becomes the apex of the cone and the radius of the circle sector becomes the slant length of the open cone.
a Consider the following sector:
i Find the area of this circle sector (and therefore the surface area of the open cone made from this sector). Give your answer to one decimal place.
ii Hence, find the radius of the open cone made from this sector correct to one decimal place.
b A circle sector with radius r and interior angle θ is made into an open cone.
i Find the value of θ if a radius of the cone is half the radius of the sector.
ii Find the value of θ if a radius of the cone is one third the radius of the sector.
iii Find the radius of the cone in terms of r and θ
15 Rhonda is creating a square-based right pyramid with a 30 cm by 30 cm square sheet of paper. She draws the following dotted lines on the paper.
a Find the surface area of the pyramid created by Rhonda when she cuts along the dotted lines and folds the triangles up.
b Find the fraction of the square paper that goes to waste.
c Rhonda’s friend Daniele suggests a different way to cut the paper in order to create a taller right pyramid with the same square base. She draws her design on another piece of paper, as shown below.
i Use Pythagoras’ theorem to find the height of the triangles that become the slanted faces of the pyramid. Give your answer correct to one decimal place.
ii Hence, find the surface area of the pyramid that is created by cutting the dotted lines this way. Give your answer correct to one decimal place.
iii Find the percentage of the square paper that goes to waste with this design. Give your answer correct to one decimal place.
By the end of this topic you will be able to ... ✔ calculate the volume of pyramids and cones.
• The volume of any pyramid (oblique or right) can be found using the formula V = 1 3Ah, where A is the area of the base and h is the perpendicular height of the pyramid.
• The volume of a cone (oblique or right) can be found using V = 1 3πr 2h, where r is the radius of the base and h is the perpendicular height of the cone.
Inter-year links
Year 7 7F 3D objects
Year 9 6D Volume of composite solids
Key content video
Find the volume of this pyramid.
1 Identify the shape of the base of the pyramid and calculate the area of the base.
h r
2 Convert the measurement given for the perpendicular height to the same units as the sides of the base.
3 Substitute the values of the perpendicular height and the area of the base into the formula for the volume of a pyramid.
4 Calculate the result and include the appropriate unit.
WRITE
The pyramid has a rectangular base.
l = 20 cm b = 12 cm
A = lb = 20 × 12 = 240 cm2
h = 150 mm = 15 cm
V = 1 3Ah = 1 3 × 240 × 15 = 1200 cm3
Calculate the volume of this cone, correct to one decimal place.
30 cm
1 Calculate the value of the cone’s perpendicular height by using Pythagoras’ theorem.
11 cm
WRITE
h2 + 112 = 302
h2 = 900 – 121
h
2 = 779
h = √779
2 Substitute the values of the radius of the base and the perpendicular height into the formula for the volume of a cone.
3 Write the answer correct to one decimal place and include the appropriate unit.
r = 11 cm
V = 1 3πr 2h
= 1 3 × π × 112 × √779
= 3536.573...
≈ 3536.6 cm3
✔ If the perpendicular height of a pyramid or cone is not given, you may need to calculate it using Pythagoras’ theorem.
✔ The formulas for the volume of a pyramid or a cone are essentially the same, with πr2 representing the area of the base of a cone.
1 Calculate the volume of each of these pyramids. Give your answers correct to one decimal place where appropriate.
2 Find the volume of each of these pyramids by first calculating the area of the base. Give your answers correct to one decimal place where appropriate.
3 The Great Pyramid of Cheops in Giza, Egypt, can be approximated by a square-based pyramid. Its perpendicular height is approximately 147 m and the length of its base is approximately 230 m. Calculate its volume.
4 Find the volume of each of these cones. Give your answers correct to one decimal place.
5 A traffic cone has a base with a diameter of 270 mm and it is 850 mm tall. Find the volume of the cone, giving your answer in cubic centimetres correct to one decimal place.
6 Calculate the volume of this hexagonal pyramid correct to one decimal place. 70 mm
7 a Use Pythagoras’ theorem to find the height of the triangular base of this tetrahedron. Then find the area of the base. Give your answers correct to one decimal place.
b Use your answer for part a to help you find the volume of the tetrahedron. Give your answer in:
i cubic centimetres
ii cubic millimetres.
c Write a formula for finding the volume of a tetrahedron.
8 a If a cone has a volume of 89.8 cm3 and the radius of its base is 3.5 cm, find the height of the cone correct to one decimal place.
b If a second cone has the same volume as the cone from part a, but has a height of 10 cm, find the radius of this cone correct to one decimal place.
9 A vase has the shape of an open inverted cone sitting in a frame. The circular top of the vase has a radius of 12 cm and the height of the vase is 46 cm.
If the vase is filled to three-quarters of its capacity with water, find how much water, in litres, that the vase contains. Give your answer correct to one decimal place.
10 a Calculate the surface area of this composite solid, correct to one decimal place.
b Which unknown dimension is also needed to calculate the volume of this solid?
c Use Pythagoras’ theorem to find the unknown length. Calculate the volume of the solid. Give your answers correct to one decimal place.
11 A cone has a base with a radius of 3 cm. The cone is 5 cm tall.
a Calculate the volume of the cone correct to one decimal place.
b Double the radius of the cone.
i What is the new radius?
ii If the height does not change, calculate the volume of the new cone correct to one decimal place.
c Double the radius once again. Using the same height, find the volume of the new cone correct to one decimal place.
d Compare your answers for parts a, b and c. Describe what happens to the volume of a cone if the radius is doubled and the height remains the same.
e This time, double the height of the original cone and leave the radius unchanged. Calculate the volume of the cone if the radius of the base is still 3 cm but the cone is now 10 cm tall. Give your answer correct to one decimal place.
f Double the height of the cone again, but do not change the radius. Calculate the volume of this new cone correct to one decimal place.
g Can you predict the volume of the cone if the height is doubled once again but the radius of the base remains unchanged? Calculate the volume to see if your prediction is correct.
h Write a sentence describing the effect on the volume of a cone when the height is doubled but the radius remains the same.
i Why is the volume increased by a different factor when the radius is doubled compared to when the height is doubled?
12 This pyramid framework is 8 cm tall, has a rectangular base with a length of 9 cm and a breadth of 5 cm. A coin of diameter 3 cm and thickness 0.2 cm sits in the centre of the pyramid. What percentage of the space within the pyramid does the coin occupy?
Give your answer correct to one decimal place.
13 A cone of height 40 cm and radius 12 cm is placed inside a square-based pyramid. The pyramid has a height of 40 cm and the edges of its square are 24 cm. This means that the pyramid is just big enough to contain the cone.
a Find the volume of the cone correct to one decimal place.
b Find the volume of the pyramid correct to one decimal place.
c Hence, find the volume of empty space between the pyramid and the cone.
d Find the exact ratio of the space in the pyramid taken up by the cone to the space in the pyramid that is empty.
14 A paper cup is in the shape of an open cone. The radius of the cone is 5 cm and the height of the cone is 10 cm.
a Find the volume of the cone correct to one decimal place.
b Find the height of the water in the cone when the cup is half full. Give your answer correct to one decimal place.
15 The tip or apex of a square pyramid is cut off parallel to the base and is removed. The object remaining is called a truncated pyramid.
a What shapes are the faces of the truncated pyramid?
b Calculate the volume of the square pyramid before the tip has been removed.
c Calculate the volume of the smaller pyramid that has been cut off correct to one decimal place.
d Use your answers to parts b and c to calculate the volume of the truncated pyramid on the right.
16 The base of a statue was cut from a concrete block formed as a square-based pyramid. If the height of the original block was 3.1 m before it was truncated, how much concrete was needed to construct this base? Give your answer in cubic metres correct to one decimal place.
17 A drinking glass is 120 mm tall and is in the shape of a truncated cone. The base of the glass has a diameter of 55 mm and the top of the glass has a diameter of 75 mm
a If the height of the conical piece of glass used to construct the drinking glass was 192 mm, calculate the volume of the drinking glass correct to one decimal place.
b How many millilitres of iced tea can this glass hold when it is filled to its brim? Give your answer correct to the nearest whole number.
18 It is possible to write a formula for finding the volume of a truncated square-based pyramid and a truncated cone. Use your answers to questions 15–17 to assist you to find these formulas.
By the end of this topic you will be able to ... ✔ calculate the surface area and volume of spheres.
Inter-year links
Year 7 8B Circumference of a circle
Year 8 8D Area of a circle
Year 9 6B Surface area of cylinders
• A sphere is a round 3D object, with every point on its surface equidistant from its centre.
The size of a sphere is determined by its radius (or diameter).
• The formula for the surface area of a sphere is:
SA = 4πr 2
• The formula for the volume of a sphere is:
V = 4 3πr 3
• A hemisphere is half of a sphere.
In this book, closed hemispheres are drawn with a shaded circular face.
Calculate the surface area of a sphere with a radius of 3 cm, correct to one decimal place.
THINK
1 Substitute the length of the radius into the formula for the surface area of a sphere.
2 Calculate the result, using the π key on your calculator.
3 Write your answer correct to one decimal place and include the appropriate unit.
Calculate the volume of this sphere correct to one decimal place.
18 cm
THINK
1 Calculate the length of the radius of the sphere.
2 Substitute the length of the radius into the formula for the volume of a sphere.
3 Calculate the result using the π key on your calculator.
4 Write the answer correct to one decimal place and include the appropriate unit.
Calculate the radius of a sphere that has a volume of 500 cm3. Give your answer correct to one decimal place.
THINK
1 Substitute the given volume into the formula for the volume of a sphere.
2 Rearrange the equation so r3 is alone on one side of the equation.
3 Use a calculator to find r
4 Write the answer correct to one decimal place and include the appropriate unit.
✔ When working with surface areas and volumes, always remember to include units with your final answers.
1 Calculate the surface area of each of these spheres correct to one decimal place.
2 Calculate the surface area of each of these spheres by first calculating the radius. Give your answers correct to one decimal place.
3 Calculate the volume of each sphere in question 1 correct to one decimal place.
4 Calculate the volume of each sphere in question 2 correct to one decimal place.
5 Calculate the radius of a sphere that has a volume of 5575 cm3. Give your answer correct to the nearest cm.
6 a Find the radius of a sphere that has a surface area of 144π cm2.
b Find the radius of a sphere that has a volume of 972π cm3
c Find the radius of a sphere that has a surface area of 2124 cm2
d Find the diameter of a hemisphere that has a volume of 1072 cm3.
7 Consider the following solid sphere on the right.
a Find the surface area of the sphere, as an exact value.
b The sphere is cut exactly in half, as shown below.
i Find the area of the circular face, as an exact value.
ii Find the area of the curved surface, as an exact value.
iii Hence, find the total surface area of the solid hemisphere, as an exact value.
8 For each hemisphere below, calculate:
i its outer surface area correct to one decimal place
ii its volume correct to one decimal place.
Hint: First determine whether each hemisphere is open or closed. In this book closed hemispheres are drawn with a shaded circular face.
9 When inflated, the beach ball below has a diameter of 76 cm.
a Calculate its surface area correct to one decimal place.
b Find the volume of air, correct to one decimal place, that it can hold.
10 a The volume of a fully inflated volleyball is 0.15 m3. Find the radius of the volleyball, to the nearest centimetre.
b The surface area of a tennis ball is approximately 564 cm2. Find the radius of the tennis ball correct to one decimal place.
11 The approximate diameters of the planets in our solar system are listed in the table below.
a Copy and complete the table. Write the values in scientific notation correct to two significant figures (a × 10m where 1 ≤ a < 10 and m is an integer).
b Approximately how many times greater is Earth’s volume compared to the volume of Mercury?
c Approximately how many times greater is Jupiter’s surface area compared to Earth’s surface area?
d Approximately 71% of Earth’s surface is covered by water. To the nearest square kilometre, how much of Earth’s surface area is covered by water?
12 The rind of a piece of fruit, like an orange, is the outer layer, whether or not it is usually eaten and can be peeled off to reveal the edible inner part of the fruit. The total surface area of the rind of this blood orange is 116.9 cm
a What is the diameter of the orange correct to one decimal place?
b The thickness of the rind is approximately 1 cm. What volume does the edible part of the orange occupy to the nearest cubic centimetre.
13 A balloon is inflated to become a sphere with a radius of 9.5 cm.
a What is the volume of the balloon correct to one decimal place?
b Two more balloons are inflated to exactly the same size as the first balloon. What is the total volume of the three balloons correct to one decimal place?
c When objects are filled with liquid or gas, you can refer to their capacity rather than volume. What is the capacity of the three balloons, in litres? Give your answer correct to one decimal place.
d If the rate that the balloons can be inflated is 2.5 L/s, how long will it take to inflate all three balloons? Give your answer correct to one decimal place.
e What assumptions do you need to make when answering part d?
Tennis balls are sold in canisters. Each canister holds four tennis balls.
a If each tennis ball has a diameter of 6.8 cm, calculate the volume of one tennis ball correct to one decimal place.
b What is the combined volume of all four tennis balls correct to one decimal place?
c Determine the dimensions of the canister and find its volume correct to one decimal place. Assume that the tennis balls fit perfectly into the canister.
d How much space in the canister is not taken up by tennis balls? Give your answer correct to one decimal place.
e If the tennis balls were to be sold in a container with the same dimensions but in the shape of a rectangular prism instead of a cylinder, without any further calculations, would there be more or less unused space?
15 Rebecca buys an ice-cream cone. As well as the ice cream on top of the cone, the cone is completely filled with ice cream. Calculate how much ice cream Rebecca has in total if the top forms a perfect hemisphere. Give your answer to the nearest millilitre.
17 The bricks shown are all equal in size and measure 23 cm by 11.5 cm by 7.5 cm. The holes on each brick are hemispheres and the diameter of each hole is 2.5 cm. Calculate the volume of clay required to make the five bricks. Give your answer to the nearest cm
18 a A ping pong ball has a diameter of 40 mm. Find its v
b Density is defined as the mass per unit volume and can be found using the formula:
If the mass of the ping pong ball is 2.7 g, calculate its density in g/cm . Give your answer correct to two decimal places.
19 A sphere can be approximated using many right square pyramids, all with equal base lengths, and slant heights equal to the radius of the sphere. The accuracy of the approximation increases as the number of pyramids increases. In this question, you will derive the formula for the surface area of a sphere.
a Suppose your approximation uses n pyramids with base area A. Write an expression for the volume of the sphere in terms of A, n and r, where r is the radius of the sphere.
b Equate your expression from part a with the usual formula for the volume of a sphere and simplify to find an expression equal to nA
c Why does your answer to part b give the surface area of the sphere?
d Why doesn’t this method work for small values of n (like n = 6)?
20 a A solid sphere of radius 10 cm is placed inside the smallest cube that can contain it.
i Find the side length of the cube.
ii Find the volume of empty space in the cube correct to one decimal place.
b A solid cube of side length 10 cm is enclosed in the smallest sphere that can contain it.
i Find the side length of the sphere correct to one decimal place.
ii Find the volume of empty space in the sphere correct to one decimal place.
Check
The SA is the sum of the areas of all the faces.
Area of a composite shape
1 Split the shape into individual parts.
2 Calculate any missing dimensions.
3 Calculate the areas of the individual parts.
4 Add or subtract the areas to nd the total area required.
V = Ah, where A is the area of the base and h is the perpendicular height of the pyramid.
• The surface area (SA) of a 3D object is the total area of its outer surface. This is the sum of the areas of the faces (or surfaces) of that object.
• For any prism, V = Ah, where A is the area of the base and h is the height of the prism.
The following key terms are used in this chapter:
•
• cylinder
• face
Chapter review quiz
Take the chapter review quiz to assess your knowledge of this chapter.
• pyramid
• quadrant
Test your knowledge of this topic by working individually or in teams.
• surface area
• tetrahedron
• circumference
•
• cone
• conversion factor
• cross-section
• hemisphere
• net
• oblique pyramid
• perpendicular
• prism
1 Why is a cylinder not a prism?
• regular tetrahedron
• right cone
• right pyramid
• sector
• sphere
• uniform
• volume
2 Draw a diagram to show the conversion factors for converting between mm2, cm2, m2 and km2.
3 Complete the following sentences using words from the key term list.
a The of a 3D object is found by summing the areas of each individual
b The of an object is how much space it takes up. The is how much liquid the object can hold.
4 Explain what it means for a 3D object to have a uniform cross-section. Give examples of objects with and without a uniform cross-section.
5 Provide a definition in your own words for the following key terms.
a regular tetrahedron
b right cone
1 Which of the following is equivalent to 300 cm2?
2 What is the SA of a cube with side lengths of 8 cm?
c sphere
3 A closed cylinder has a base diameter of 15 cm and a height of 15 cm. Which of the following is closest to the surface area of the cylinder?
4 A cylinder has a base diameter of 6 cm and a height of 20 cm. Which of the following is closest to the volume of the cylinder?
5 A solid wooden cylinder has a base with radius 10 cm and is 10 cm tall. It has a cylinder of radius 2 cm drilled through its centre, from top to base. What is the volume of timber remaining (in cubic centimetres)?
A 640π cm3 B 1000π cm3 C 960π cm3 D 80π cm3 E 40π cm3
6 A square-based pyramid has a surface area of 224 cm2. If the length of the base is 8 cm, what is the height of each triangular face?
A 56 cm B 14 cm C 10.5 cm D 10 cm E 3.5 cm
7 How many triangular faces does a tetrahedron have?
A 1 B 3 C 4 D 5 E 6
8 A square-based pyramid has the same base length and height as a cube with side lengths of 6 cm. What is the volume of the pyramid?
A 216 cm3 B 72 cm3 C 12 cm3 D 108 cm3 E 144 cm3
9 Which of the following is closest to the volume of a cone with a diameter of 2.5 cm and a height of 10 cm?
A 16 cm3 B 25 cm3 C 50 cm3 D 65 cm3 E 200 cm3
10 Which of the following is closest to the total exposed surface area if a solid sphere with a diameter of 25 cm is sliced in half?
1 A flat metal washer is in the form of an annulus. The radius of the inner ring of the washer is 1 cm, while the outer ring has a radius of 1.5 cm. What is the area of the annulus correct to one decimal place?
2 Determine the area of each of these composite shapes. Give your answers correct to one decimal place.
3 The base of a triangular prism is a right-angled triangle with a base length of 25 mm and a height of 18 mm. The height of the prism is 30 mm. What is the SA of the prism?
4 A set of three wooden cubes was made for a playground. The smallest cube had a side length of 40 cm. The next sized cube had side lengths twice as long as the smallest cube, while the side lengths of the largest cube were twice as long as those of the middle cube.
a Calculate the surface area of each cube.
b If 1 L of paint covers an area of 15 m2, and the paint is only available in 5 L cans, how many cans of paint would be required to paint all three cubes?
5 A can of baked beans has a cylindrical shape with a base diameter of 7 cm and a height of 10 cm. What is the capacity of the can correct to the nearest millilitre?
6 Calculate the volume of each cube in question 4
7 An inground pool in the shape of the right prism shown is to be installed. The prism is not drawn to scale. Calculate the volume of soil which needs to be excavated so that the pool fits exactly into the hole.
8 A square-based pyramid has a base with sides 5 cm in length. The height of each triangular face is 20 cm. Calculate the surface area of the pyramid.
9 The diameter of a closed cone is 10 cm and the cone’s height is 16 cm. Calculate the surface area of the cone correct to one decimal place.
10 A square-based pyramid has a base with sides 30 cm. The triangular faces are equilateral. Find the volume of the pyramid.
11 A sphere fits neatly inside a cylinder with a base radius of 8 cm. Find the volume of the cylinder and the sphere in terms of π
12 a Calculate the total surface area, correct to one decimal place where appropriate, of:
i a solid hemisphere with a diameter of 40 cm
ii a cube with sides 40 cm in length.
b The hemisphere is placed on top of the cube. What is the total surface area of this object?
a Name the solids in order, from left to right.
b The cube has side lengths of 3 cm. Calculate its:
i surface area
ii volume.
c The rectangular prism has a base the same size as the cube, and is twice the height of the cube.
i Compare the surface area of the rectangular prism with that of the cube.
ii Compare the volume of the rectangular prism with that of the cube.
d The triangular prism is the same shape as the rectangular prism cut vertically along the diagonal of its base.
i Draw a labelled diagram showing the shape of the triangular prism’s base.
ii Calculate the surface area of the triangular prism correct to one decimal place.
iii Is the SA of the triangular prism half that of the rectangular prism that you calculated in part c? Explain.
iv Compare the volumes of the triangular prism and the rectangular prism.
e The base of the cylinder has a diameter the same length as the side of the cube, and the cylinder is the same height as the rectangular prism. Calculate, correct to one decimal place, the cylinder’s:
i surface area
ii volume.
f Create two tables listing the solids in parts b–e in increasing order of:
i surface area
ii volume.
g The square base of the pyramid is the same shape as that of the base of the cube. Its height is the same as that of the rectangular prism. Calculate, correct to one decimal place, the pyramid’s:
i surface area
ii volume.
h The cone has the same base diameter and height as the cylinder. Calculate, correct to one decimal place, the cone’s:
i surface area
ii volume.
i The sphere has the same diameter as the cone. Calculate, correct to one decimal place, the sphere’s:
i surface area
ii volume.
j Using your answers for parts g, h and i, update your tables from part f
2 A regular icosahedron is a polyhedron (a 3D object with polygonal faces) with 20 equilateral triangle faces.
a Find the area of an equilateral triangle with side length a.
b Find the surface area of a regular icosahedron where each equilateral triangle face has side length a.
A regular icosahedron can be divided into 20 congruent tetrahedrons. The base of each tetrahedron is one of the faces of the icosahedron, and the height of each tetrahedron is √3 (5 + √3 ) 12 a
c Determine the volume of a regular icosahedron in terms of a in exact form and as an approximation correct to three decimal places.
The circumscribed sphere of a regular icosahedron is the sphere which touches all vertices of the icosahedron and has radius with length r = a 4 √10 + 2√5
d Determine the surface area and volume of the circumscribed sphere of a regular icosahedron with side length a correct to three decimal places.
e For the case a = 1, calculate the percentage of the circumscribed sphere which is filled by the icosahedron correct to two decimal places.
A truncated icosahedron is obtained by cutting off each vertex of a regular icosahedron so that each resulting edge is one-third of the length of the edges in the original icosahedron. There are 12 regular pentagonal faces and 20 regular hexagonal faces in a truncated icosahedron. Soccer balls are based on the shape of a truncated icosahedron, where the pentagons are usually coloured black and the hexagons are usually white. The area of a regular pentagon with side length a is approximately A = 1.720a2
The area of a regular hexagon with side length a is A = 3√3 2 a2
f Determine the surface area of a truncated icosahedron with side length a 3 in terms of a correct to three decimal places.
The formula for the volume of a truncated icosahedron with side length a is V = 125 + 43√5 4 a3
g Explain why the volume of a truncated icosahedron is less than the volume of the icosahedron it is constructed from.
h Determine the volume of a truncated icosahedron with side length a/3 in terms of a correct to three decimal places.
The radius of the circumscribed sphere of a truncated icosahedron with side length a is r = a 4 √58 + 18√5
i Determine the surface area and volume of the circumscribed sphere of a truncated icosahedron with side length a 3 correct to three decimal places.
j For the case a =1, use your answers to parts h and i to calculate the percentage of the circumscribed sphere which is filled by the truncated icosahedron correct to two decimal places.
k Suggest one reason why soccer balls are in the shape of a truncated isocahedron instead of a regular icosahedron.
Now that you have completed this chapter, reflect on your ability to do the following.
I can do this
Calculate the area of composite shapes
Calculate the area of sectors
Calculate the surface area of prisms and cylinders
Calculate the surface area of composite solids consisting of prisms and cylinders
Calculate the volume of prisms and cylinders
Calculate the volume of composite solids consisting of prisms and cylinders
Calculate the surface area of right prisms and right cones
Calculate the volume of pyramids and cones
Calculate the surface area and volume of spheres
I need to review this
Go back to Topic 8A
Area review
Go back to Topic 8B
Surface area review
Go back to Topic 8C
Volume review
Go back to Topic 8D
Surface area of pyramids and cones
Go back to Topic 8E
Volume of pyramids and cones
Go back to Topic 8F
Surface area and volume of spheres
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M: 0411 759 659
Paul McCallum
E: paul.mccallum@oup.com
M: 0423 241 455
Queensland
Chris Lowcock
E: christopher.lowcock@oup.com
M: 0428 949 935
Melissa Wearne
E: melissa.wearne@oup.com
M: 0447 557 931
New South Wales
Caly James
E: caly.james@oup.com
M: 0413 745 855
Lisa Cobb
E: lisa.cobb@oup.com
M: 0411 759 608
Tasmania
Caitlin Hansford
E: caitlin.hansford@oup.com
M: 0411 759 659
Australian Capital Territory
Caly James
E: caly.james@oup.com
M: 0413 745 855