Sample Chapter - NSW Year 8 v2 - 10. Area

Big ideas

Geometric figures have properties and relationships that allow us to calculate and analyse their sizes and dimensions.

Chapter outline

Idea summary

Techniques we can use to estimate area:

• Use a mental image of common objects of size 1 mm2, 1 cm2, 1 m2 and/or 1 km2 and try to picture how many it would take to cover the object.

• For shapes that are rectangular (or close to rectangular), estimate the length and width of the object and then multiply: Area = length × width.

• If a picture of the item is available, you could draw a scale grid over it and estimate by counting the number of squares needed to cover the item.

Convert units of area

When we convert units of area we are changing the size of the squares that the area is measured in.

Interactive exploration

Explore online to answer the questions mathspace.co

Use the interactive exploration in 10.01 to answer these questions.

1. Check ‘Show Grid’ and cm2 to mm2. How many mm² in 1 cm2?

2. Check m2 to mm2. How many mm2 in 1 m2?

3. Check m2 to cm2. How many cm2 in 1 m2?

4. Check km2 to m2. How many m2 in 1 km2?

To convert units of area, such as from square centimetres (cm2) to square millimetres (mm2), consider the square with a side length of 1 cm. How many millimetres is each side length?

There are 10 mm in each 1 cm, so the square now becomes a square with side lengths of 10 mm each. Which has an area of 10 × 10 = 100 mm2?

When converting between units of area:

• Multiply if converting to a smaller unit — more smaller squares will be needed to cover the same area.

• Divide if converting to a larger unit — fewer larger squares will be needed to cover the same area.

Example 5

Convert 6 m2 to cm2

Create a strategy

Sketch a square with side length of 1 m. We want to convert to centimetres, how many centimetres will each side length be and work out how many square centimetres is in 1 square metre.

Apply the idea

6 m2 = 6 × (100 × 100) Convert the square metre into centimetres = 60 000 cm2

Evaluate

Idea summary

When converting between units of area:

• Multiply if converting to a smaller unit — more smaller squares will be needed to cover the same area.

• Divide if converting to a larger unit — fewer larger squares will be needed to cover the same area.

Practice questions

Choose your question path

Explorer

1a-d, 2, 5–8, 10ab, 11, 12, 14–18

Adventurer

1e-h, 2, 3, 5–8, 10bc, 11–13, 15–21

Trailblazer

1i-l, 2–4, 6, 7, 9, 10cd, 11–13, 15–23

Understanding

1 From these units of measure,

• Hectares

• Square centimetres

• Square metres

• Square kilometres choose the most appropriate unit for these areas:

a Postage stamp b Bathroom wall c Paper d Cabinet

e City f Television g Kitchen h Farm

i Netball court j Table k National park l Door

2 Name an object that has an area of approximately 1 m2

3 Name an object that has an area of approximately 500 cm2.

Ex 1, 2

Fluency

4 Find the area of a square of side 290 cm, in square metres.

5 Estimate the area of the Mathspace logo on the box.

6 This garden bed has three rows of four plants. Each plant is spaced about 24 cm apart, and sits 12 cm away from the edge of the garden bed.

Estimate the area of the garden bed in square centimetres.

7 Each square on the grid has an area of 4 mm2. Estimate the area of the curved shape.

8 Estimate the area of the shape shown if each axis is measured in metres.

9 Estimate the area of the butterfly if each axis is measured in millimetres.

10

Estimate the area:

a A 10 cm2

B 50 cm2

C 1 m2

D 100 mm2

c A 7000 cm2

B 6000 mm2

C 7 m2

D 400 mm2

11 Convert:

a 3 m2 to square centimetres

c 0.56 km2 to square metres

e 10.4 ha to square metres

g 3.5 m2 to square millilmetres

i 16.4 ha to square kilometres

k 0.38 km2 to hectares

Reasoning

b A 200 cm2

B 55 000 mm2

C 6 m2

D 6000 cm2

d A 7000 mm2

B 2 m2

C 70 cm2

D 2000 mm2

b 7 m2 to square centimetres

d 6 ha to square metres

f 20 000 m2 to hectares

h 27.8 cm2 to square millilmetres

j 273 mm2 to square metres

l 0.101 km2 to square millimetres

12 There are 100 cm in 1 m but to convert from square metres to square centimetres we must multiply by 10 000. Explain why this is correct.

13 Determine whether these statements are true or false. Explain your reasoning.

a The unit of estimation must always be the same as the unit of measurement used to find the exact area.

b The area of a square is equal to the perimeter of the square divided by 4.

c The area of a rectangle with length 5 cm and width 2 cm is equal to the area of a rectangle with length 2 cm and width 5 cm.

d The area of a shape can never be greater than its perimeter.

e When estimating area, it is always best to round the measurements to the nearest whole number.

14 Why is it a good idea to estimate the area of a shape before calculating it exactly?

15 A hectare is approximately the size of an athletics arena. Why are blocks of land, farms and large properties measured in hectares?

Problem-solving

16 These road signs are pictured with relative sizings. If the GIVE WAY sign is an equilateral triangle with a side length of 75 cm, estimate the area of the BICYCLES EXCEPTED sign.

17 A park has an area measuring 19 acres. Given that 1 acre is approximately 4000 m2, find the area of the park in square metres.

18 A sample of rectangular farms around Australia were measured and their dimensions, in metres, are recorded in the table:

a Find the area of each farm in square metres.

b Which farms have an area of more than 1 ha?

c Which farms have an area of less than 1 ha?

d Which farm has an area of exactly 1 ha?

19 A property covers an area of 7.22 ha. Given that 1 ha is 10 000 m2, calculate:

a The area of the property in square metres.

b The area of the property in acres, given that 1 acre is approximately 4000 m2

20 A large cattle station covers an area of 320 acres. Given that 1 acre is approximately 0.4 ha, and 1 ha is 10 000 m2, calculate:

a The area of the cattle station in hectares.

b The area of the cattle station in square kilometres.

21 A farm covers an area of 51 000 m2. Given that 1 ha is 10 000 m2, and 1 acre is approximately 4000 m2, calculate:

a The area of the farm in hectares.

b The area of the farm in acres. Round your answer to two decimal places.

22 Bob owns a paddock with an area of 290 acres. He wants to advertise his eggs as free range, with no more than 400 chickens per hectare. Given that 1 acre is approximately 0.4 ha, find the maximum number of chickens that Bob can keep in his paddock.

23 A farm covers an area measuring 340 acres. Given that 1 acre is approximately 0.4 ha, and 1 ha is 10 000 m2, calculate:

a The area of the farm in hectares.

b The area of the farm in square metres.

10.02 Area of rectangles, triangles and parallelograms

After this lesson, you will be able to…

• calculate the area of rectangles and parallelograms using the appropriate formula.

• find the missing dimension of a rectangle or parallelogram given its area and one of its dimensions.

• apply knowledge of area calculation to solve real-world problems that involve rectangles and parallelograms.

Area formulas

The area of a rectangle is given by Area = length × width, or A = l × w

The area of a square is given by Area = side × side, or A = s × s = s 2

The area of a triangle is given by

The area of a parallelogram is given by Area = base × height, or A = b × h

Example 1

Find the area of a rectangle whose length is 11 cm and width is 5 cm.

Create a strategy

Use the area of a rectangle formula.

Apply the idea

Area = l × w

= 11 × 5

= 55 cm2

Example 2

Start with the formula of the area of a rectangle

Substitute l = 11 and w = 5

Evaluate

Find the area of the given triangle with base length 10 m and perpendicular height 8 m.

m

m

Create a strategy

Use the formula for the area of a triangle: A = × b × h

Apply the idea

For this triangle, the base is b = 10 m and the height is h = 8 m.

Substitute b and h

Evaluate

Example 3

Find the area of a parallelogram whose base is 15 cm and height is 7 cm.

Create a strategy

Use the area of a parallelogram formula.

Apply the idea

A = b × h

= 15 × 7

= 105 cm2

Start with the formula for the area of a parallelogram

Substitute b = 15 and h = 7

Evaluate

Idea summary

The area of a rectangle is given by:

A = l × w

A is the area of the rectangle l is the length of the rectangle w is the width of the rectangle

The area of a square is given by:

A = s × s

A is the area of the square s is the side length of the square

The area of a triangle is given by:

A is the area of the triangle b is the base of the triangle h is the height of the triangle

The area of a parallelogram is given by:

A = b × h

A is the area of the parallelogram b is the base length of the parallelogram h is the perpendicular height

Find dimensions from the area

To find a missing dimension, we substitute the area and the known dimension into the area formula and solve for the unknown.

The parallelogram shown has an area of 792 m2, and a base length of 88 m. How can we determine the perpendicular height? h

From the formula, we know that Area = b × h

Substitute known values for A and b:

So the height of the parallelogram is 9 m.

Example 4

A rectangular park has an area of 36 m2. If the park is 4 m wide, how long is it?

Create a strategy

Use the area of a rectangle formula.

Apply the idea

Start with the formula for the area of a rectangle

Substitute A = 36 and w = 4

Divide both sides by 4

Evaluate

Idea summary

Find a missing dimension of a rectangle, square or parallelogram by substituting the area and given dimension into the area formula, and then solve for the unknown variable.

Practice questions

Choose your question path

Explorer

1–3, 4ac, 5, 6ace, 7, 10, 11ace, 12a, 13ace, 14ac, 15, 17ac, 18ac, 19, 20, 23, 25, 27

Understanding

1 For the image shown:

a The length is a and the width is b.

Adventurer

2, 3, 4bd, 5, 6bdf, 8, 10cd, 11bdf, 12bc, 13bdf, 14bd, 16, 17bd, 18bd, 20, 21, 24, 26, 27

Trailblazer

2, 3, 4bd, 5, 6ef, 8, 9, 10cd, 11ef, 12d, 13ef, 14d, 16, 17bd, 18bd, 20–22, 24, 26–29

Explain how we would work out the area of the rectangle.

b Write an expression for the area of the rectangle.

c What would the units be for the area?

d If the lengths had been in metres instead of centimetres, what would the units be for the area? b cm a cm

2 Casper, Ollie and Stella are working out the area for the rectangle shown, given that x = 2 and y = 20.

a Stella suggests multiplying 2 and 20. Explain why Stella is incorrect.

b Ollie plans to change the length into centimetres, so he will multiply 200 and 20. i Is Ollie using a valid method?

ii What would be the units for Ollie’s answer?

c Casper intends to change the width into metres, so he will multiply 2 and 0.2 i Is Casper using a valid method?

ii What would be the units for Casper’s answer?

3 Alex and Beth are calculating the area of the square shown, where a = 4.

a Alex says that a square is a special case of a rectangle, so he can work out the area by calculating 4 × 4. Is Alex correct?

b Beth says that the formula for the area of a square is A = a2, so she calculates 42. Is Beth correct?

c After realising that they both got an answer of 16, they conclude that they could use either method. Are they correct?

d What would be the units for their answer?

4 Identify the base and height of the triangle:

5 A parallelogram with a base of 9 cm and a height of 6 cm is formed into a rectangle:

a Find the length of the rectangle.

b Find the width of the rectangle.

c Calculate the area of the rectangle.

d Explain why the area of the parallelogram and the rectangle are the same.

e Find the base of the parallelogram.

f Find the height of the parallelogram.

g With reference to the diagram, explain why the formula for the area of a parallelogram is A = bh

Fluency

6 Calculate the area of the rectangles:

a length = 7 cm, width = 4 cm

c length = 15 m, width = 6 m

e length = 5 m, width = 40 cm

b length = 12 mm, width = 5 mm

d length = 6 km, width = 3 km

f length = 12 cm, width = 6 mm i in cm2 ii in m2 i in cm2 ii in mm2

7 A square has a perimeter of 20 cm. Calculate the area of the square.

8 A rectangle has a perimeter of 20 cm and a length of 8 cm. Calculate the area of the rectangle.

9 A rectangle has an area of 20 cm2. The length and width are integer values, and the length is greater than the width. Write down the dimensions of three different rectangles that satisfy these conditions.

10 Find the area:

11 Calculate the area of the parallelograms:

a base = 9 m, height = 7 m

b base = 10 mm, height = 5 mm

c base = 20 cm, height = 3 cm

d base = 14 m, height = 5 m

e base = 8 km, height = 400 m f

= 9 m,

= 4 cm i

12 Find the width of the rectangles, given the area and the length.

a area = 14 m2, length = 7 m

c area = 36 mm2, length = 9 mm

b area = 28 cm2, length = 14 cm

d area = 50 cm2, length = 12.5 cm

13 Find the area of the triangle that has the dimensions:

a Height = 10 cm and base = 7 cm b Height = 5 m and base = 3 m

c Height = 6 mm and base = 4 mm d Height = 2 cm and base = 8 cm

e Base = 15 cm and height = 80 mm f Base = 6.2 m and height = 180 cm

14 Find the height of the parallelograms, given the area and the base length.

a area = 40 cm2, base = 8 cm b area = 56 m2, base = 7 m

c area = 60 mm2, base = 15 mm d area = 39 km2, base = 3 km

15 Complete the table for lengths, widths and areas of rectangles.

Length (cm) 11 16

Width (cm) 9 4

Area (cm2) 54 55

16 Complete the table for bases, heights and areas of parallelograms.

Base (cm) 16 25

Height (cm) 13 5

Area (cm2) 52 80

17 Find the base measurement for the triangle given the area and height:

a Area = 24 cm2 and height = 6 cm b Area = 35 cm2 and height = 7 cm

c Area = 14 mm2 and height = 7 mm d Area = 300 mm2 and height = 3 cm

18 Find the height measurement for the triangle given the area and base:

a Area = 50 cm2 and base = 10 cm b Area = 100 cm2 and base = 25 cm

c Area = 27 mm2 and base = 3 mm d Area = 800 mm2 and base = 40 cm

Reasoning

19 a Draw a diagram showing how we can re-form this parallelogram into a rectangle.

b If the parallelogram is formed into a rectangle, find:

i The length of the rectangle ii The width of the rectangle

c Find the area of the parallelogram.

20 The dimensions of the large rectangle are double the size of the small rectangle.

a Find the value of x

b Is the area of the large rectangle double the area of the small rectangle?

c If the side lengths of a rectangle are doubled, by what factor is the area increased?

d If the side lengths of a rectangle are tripled, by what factor is the area increased?

21 Two right triangles have the same area. If one triangle has a base of 7 cm and a height of 9 cm, what are the possible dimensions of the other triangle given the base and height are whole numbers?

22 A rectangle and parallelogram each have sides of a cm and b cm. Which shape would have the greater area? Explain.

Problem-solving

23 A rectangular swimming pool measures 12 metres by 8 metres.

a What is the area of the pool in square metres?

b If the pool deck surrounding the pool is 2 metres wide, what is the total area of the pool and deck in square metres?

24 A rectangular garden measures 12 metres by 8 metres. A path runs around the outside of the garden, with a width of 1 metre. What is the area of the path in square metres?

25 The floor of a room measures 8 metres by 10 metres. What is the area of the floor in square metres if the room is in the shape of a parallelogram?

26 A farmer wants to install a fence around his rectangular field. One of the longer sides of the field is 60 metres, and the shorter side is 30 metres. He wants to add a rectangular-shaped area to the field that extends 20 metres out from the shorter side. What is the area of the rectangular-shaped area he wants to add?

27 A rectangular room has a floor that measures 4 m by 3 m. The floor is to be tiled with tiles that are a 25 cm square. How many tiles are required to completely cover the floor?

28 Amy wants to paint her bedroom walls. Two of the walls measure 4 m high and 6 m wide, while the other walls measure 4 m high and 5 m wide. There is a triangular window on one of the walls that has a base of 4 m and height of 2 m. If each litre of paint can cover an area of 8 m2, how many whole litres of paint will Amy need to buy?

29 Ryan is re-tiling his rectangular kitchen floor. The kitchen measures 3 m by 5 m, but there is an island in the centre in the shape of a right triangle with a base of 6 m and a height of 2 m. Each tile measures 30 cm by 30 cm.

a Calculate the number of tiles that Ryan needs to buy.

b If Ryan needs to account for 15% wastage, how many tiles does he need to buy?

10.03 Area of trapeziums, kites, and rhombuses

After this lesson, you will be able to…

• define and identify rhombus, kite, and trapezium.

• understand and apply the formula for the area of a rhombus.

• understand and apply the formula for the area of a kite.

• understand and apply the formula for the area of a trapezium.

• compute the area of a rhombus, kite, and trapezium given specific dimensions.

• deduce a missing dimension of a rhombus, kite, or trapezium when given the area and one other dimension.

Area of a rhombus

A rhombus is a four-sided shape with four equal sides and opposite sides parallel. It is similar to a square, but doesn’t necessarily have right angles. This means a square is a special type of rhombus, and a rhombus is a special type of parallelogram.

If we are given the base and perpendicular height, then we can use the formula for the area of a parallelogram to find the area of a rhombus.

If we are given the lengths of the diagonals, we need another method for finding the area of a rhombus.

We can use the similarities between rhombuses and rectangles to find the formula for area.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 10.03 to answer these questions.

1. What is the area of the largest rectangle that encloses the rhombus?

2. What portion of this area does the area of the rhombus compose?

3. Explain how the area of a rhombus is connected to the area of a rectangle.

We can turn a rhombus into a rectangle that has one dimension equal to the length of one of the diagonals while the other dimension will be half the other diagonal. Notice that it does not matter which diagonal we halve.

Example 1

Find the area of the rhombus.

Create a strategy

We can find the area of rhombus using the formula:

Apply the idea

We can see from the diagram that the diagonals have lengths of 4 mm and 8 mm.

Start with the formula for the area of a Rhombus

Substitute the values

Evaluate the product

Idea summary

The area of a rhombus can be found using the formula: x is the length of one diagonal y is the length of the other diagonal

Area of a kite

A kite is similar to a rhombus, but it has two pairs of equal sides rather than four equal sides. This means a rhombus is a special type of kite.

We can use the same formula for the area of a kite as the area of a rhombus.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 10.03 to answer these questions.

1. Describe a relationship between the diagonals of a kite and the sides of the rectangle you created.

2. Explain how the area of a kite is connected to the area of a rectangle.

Once again, by turning the kite into a rectangle, we can see that we end up with a rectangle that has dimensions equal to the diagonals of the kite.

Find the area of the kite.

Create a strategy

We can use the formula:

Apply the idea

Start with the formula for the area of a kite

Substitute the diagonal lengths

Evaluate

Example 2

Idea summary

The area of a kite can be found using the formula: x is the length of one diagonal y is the length of the other diagonal

Area of a trapezium

Trapeziums (or “trapezia”) have one pair of parallel sides.

We can use the area of a parallelogram to help us work out the area of a trapezium.

Interactive exploration

Explore online to answer the questions mathspace.co

Use the interactive exploration in 10.03 to answer these questions.

1. What quadrilateral do two trapeziums make?

2. Describe the relationship between the base length of a trapezium and the base lengths of the created parallelogram.

3. Explain how the area of a trapezium is connected to the area of the created parallelogram.

4 Explain how the area of a trapezium is connected to the area of a rectangle that has the same base length and height as the parallelogram.

We can turn a trapezium into a rectangle, as demonstrated in the image.

The length, l, will be the average length of the two parallel sides.

The width is 8 cm. A = l × w = 9.5 × 8 = 76 cm2

The area of a trapezium is given by: Area = × (side a + side b) × height h

Where a and b are the lengths of the parallel sides, and h is the perpendicular height.

Example 3

Find the area of the trapezium.

Create a strategy

We can find the area of trapezium using the formula:

Apply the idea

The side lengths are a = 5 cm and b = 8 cm, and the height is h = 3 cm.

Start with the formula for the area of a trapezium

Substitute the values

Evaluate the product

Idea summary

The area of a trapezium is given by:

a is a parallel side length

b is the other parallel side length h is the perpendicular height

Unknown dimensions

If we know the area of the shape and one other dimension, then we can find an unknown dimension using the appropriate area formula and solving for the unknown dimension.

Example 4

The kite has an area of 48 cm2. The length of one of its diagonals is 12 cm. k cm 12 cm

If the other diagonal has a length of k cm, solve for the value of k.

Create a strategy

Substitute the area, the known diagonal, and the variable k into the formula for the area of a kite.

Apply the idea

Start with the formula for the area of a kite

Substitute the values and k

Evaluate the product

Divide both sides by 6

Evaluate

Idea summary

We can find an unknown dimension of a quadrilateral if we know the area and the other dimension. We substitute the known values into the area formula and solve for the unknown dimension.

A summary of the area formulas for quadrilaterals:

Practice questions

Choose your question path

Explorer

Adventurer

Trailblazer 1, 2, 4, 5, 7–12, 14, 24, 25, 27, 30–32, 39 1, 2, 6, 8, 9, 11–26, 29–33, 36, 37, 39, 40 3, 15–29, 34–38, 40, 41

Understanding

1 What is a trapezium?

A A quadrilateral with all sides equal

B A quadrilateral with one pair of parallel sides

C A quadrilateral with all angles equal

D A quadrilateral with no parallel sides

2 What is a kite?

A A quadrilateral with no parallel sides

B A quadrilateral with one pair of parallel sides

C A quadrilateral with all angles equal

D A quadrilateral with all sides equal

3 Here is a rhombus and a kite. What is the difference between them?

Fluency

4 This rhombus can be split into two triangles:

a Find the area of one triangle.

b Find the area of the rhombus.

5 Find the area:

6 Determine whether the pair of values could be the diagonal lengths, x and y of a rhombus with an area of 9 m2

a x = 2 m and y = 9 m

c x = 12 m and y = 3 m

7 This kite can be split into two triangles:

a Find the area of one of the triangles.

b Find the area of the kite.

8 This kite is formed into a rectangle:

a Find the length of the rectangle.

b Find the width of the rectangle.

c Find the area of the kite.

b x = 6 m and y = 3 m

d x = 6 m and y = 6 m

9 Find the area:

10 The trapezium is split into a rectangle and a right-angled triangle:

a Find the area of the rectangle.

b Find the area of the triangle.

c Find the area of the trapezium.

11 This trapezium is formed into a rectangle:

a Find the length, l, of the rectangle.

b Find the area of the trapezium.

12 Two identical trapeziums are put together to make a rectangle:

a Find the area of the entire rectangle.

b Find the area of one trapezium.

13 Find the area:

14 Find the area:

15 For each rhombus, find the value of the variable: a A = 64 cm2 b A = 128 cm2

16 Rhombus ABCD has an area of A = 55 cm2:

Given the diagonal BD = 11 cm and AC = x cm, find the value of x.

17 Rhombus ABCD has an area of 13 cm2: If diagonal AC = 2 cm and diagonal BD = y cm, find the value of y

18 The kite has an area of 15 cm2. The length of one of its diagonals is 3 cm. Find the length of the other diagonal, k.

19 Find the value of k: a A = 48 cm2 b A = 22.5 cm2

c A = 56 cm2 d A = 137.5 cm2

20 Find the value of the variable:

a A = 42 mm2 b A = 36 cm2

c A = 20 m2 d A = 24 cm2

21 Find the value of x if the area of the trapezium is 65 cm2:

22 The area of a kite is 640 cm2 and one of the diagonals is 59 cm. If the length of the other diagonal is y cm, find the value of y, rounded to two decimal places.

23 Complete the table of the lengths of diagonal x and diagonal y for three kites that all have an area of 36 mm2:

Area (mm2) Diagonal, x (mm) Diagonal, y (mm)

24 A field has a rhombus shape with diagonals of 120 metres and 80 metres. What is the area of the field in square metres?

25 A gardener wants to create a flower bed in the shape of a rhombus. The length of one diagonal is 6 metres, and the other diagonal is 8 metres. What is the area of the flower bed in square metres?

26 A farmer wants to plant a field in the shape of a trapezium. The shorter base of the trapezium is 80 metres and the longer base is 120 metres. The height of the trapezium is 50 metres. If the farmer can plant 2500 seeds per square metre, how many seeds will he need to plant the entire field?

27 A carpenter needs to cut a piece of wood in the shape of a trapezium. The longer base of the trapezium is 60 cm and the shorter base is 40 cm. The height of the trapezium is 24 cm. How many square centimetres of wood will the carpenter need to make the piece?

28 A kite-shaped water feature has diagonals of 30 metres and 20 metres. The owner wants to place tiles on the pool floor. If each tile is a square with an area of 0.25 square metres, how many tiles are needed to cover the entire pool floor?

29 A kite-shaped playground has an area of 24 square metres. The length of one diagonal is 10 metres. What is the length of the other diagonal?

Reasoning

30 What is a trapezium? Can you explain its properties and how it differs from a parallelogram or rectangle?

31 What is a rhombus? Can you explain its properties and how it differs from a square?

32 Determine whether each statement is true or false. Justify your answer.

a The area of a trapezium is always equal to the sum of the lengths of its parallel sides multiplied by its height divided by 2.

b A rhombus is always a square.

c A rhombus is a type of kite.

d The area of a kite can be found by multiplying the lengths of its diagonals and dividing by two.

33 This rhombus is formed into a rectangle:

a Find the length of the rectangle in terms of y

b Find the width of the rectangle in terms of x.

c Find the area of the rectangle in terms of x and y

d Find the area of the rhombus in terms of x and y.

Length x y

Width

34 This kite is formed into a rectangle:

a Find the length of the rectangle in terms of y.

b Find the width of the rectangle in terms of x

c Find the area of the rectangle in terms of x and y.

d Find the area of the kite in terms of x and y

35 Justin is trying to work out the area of a rhombus, but he has been given the base and height of the shape. Gaynor says that is fine, he can just use A = bh. Is Gaynor correct? Explain.

36 Use the image to show that the area of the parallelogram is equivalent to twice the area of the trapezium.

mm

mm

mm

37 A kite is cut from a rectangular sheet as shown. Explain why it is not possible to cut a kite spanning the length of the sheet with a larger area.

Problem-solving

38 A gardener charges $2 per square metre to mow lawns.

a How much would she charge to mow a trapezoidal lawn with parallel sides 8 m and 10 m, the distance between the parallel sides being 5 m?

b How much would she charge to mow a kite-shaped lawn with diagonals 12 m and 8 m?

c She charged $120 to mow a lawn in the shape of a rhombus with one diagonal of length 10 m. What was the length of the other diagonal?

39 For the rhombus in the diagram, one of the diagonals is of length 9 cm. How long is the other diagonal?

cm

Investigate

After completing the activity, discuss the questions below with a classmate.

1. When a circle is cut into slices and rearranged, what shape(s) can be formed?

2. How does the difference between the area of the circle and area of the sectors change as the number of slices increases?

3. In general, for a circle with radius r cm, what would the area be?

Answer these questions after completing the activity.

Discussion

1. Use the formula to find the exact area of a circle with a radius of 6 cm. Then, use a calculator to approximate the area to 2 decimal places. Is the area rational or irrational?

2. Do you think the area of a circle will always be irrational? Explain your reasoning.

10.04 Area of a circle

After this lesson, you will be able to…

• understand and explain the concept of a circle’s area and how it relates to its radius.

• apply the formula for the area of a circle in various contexts.

• calculate the area of a full circle, a semicircle, a quarter circle, and three-quarters of a circle, using the given radius.

• solve problems requiring the calculation of a circle’s area, given its radius, and vice versa.

• deduce an unknown radius when given the area of a circle.

• solve real-life problems involving areas of circles and segments of circles.

• calculate the circumference of a circle given its area, applying mathematical skills to solve for radius first.

• demonstrate precision in calculation and rounding to a specified number of decimal places.

Area of a circle

Circles don’t have straight edges, but we can find an expression for the area of a circle by finding the area of a set of isosceles triangles which share a common vertex at the centre

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 10.04 to answer these questions.

1. What happens when you unravel the circle?

2. When the triangles are rearranged to form a parallelogram, what do the base and height of that rectangle represent in terms of the circle’s properties?

3. How is the formula Area = πr 2 derived?

The more triangles we use, the closer their combined area gets to the area of the circle.

We can calculate the area of a circle using the formula:

where A is the area and r is the radius of the circle.

To find the area of a semicircle (half a circle) we multiply the area by

To find the area of quarter of a circle we multiply the area by

To find the area of three-quarters of a circle we multiply the area by .

Example 1

Find the area of the circle, rounded to one decimal place.

6 cm

Create a strategy

Use the formula:

Apply the idea

A = πr 2

= π(6)2

= 113.1 cm

Area of circle = πr 2

Start with the formula for the area of a circle

Substitute r = 6

Evaluate, rounded to one decimal place

Example 2

Calculate the area of the figure, rounded to one decimal place.

2 cm

Create a strategy

The area of the figure is equivalent to three-fourths the area of the circle.

Apply the idea

Remember: Area of a circle = πr 2

A = πr 2

Start with the formula for the area of a circle

Substitute r = 2

Evaluate, rounded to one decimal place

Idea summary

We can calculate the area of a circle using the formula:

A = πr 2

A is the area

r is the radius of the circle

We multiply the area formula by the relevant fraction to find the area of a shape that is a quarter, half or three quarters of a circle.

Find unknown radius

The area formula can also be used to find the radius if we know the area of the circle.

To solve for the radius we will need to rearrange the formula for the area of a circle:

Swap the sides about the equals sign

Divide both sides by π

Take the square root of both sides

Example 3

A circle has an area of 49 mm2. What is its radius, rounded to two decimal places?

Create a strategy

Substitute the area into the formula A = πr 2

Apply the idea

Start with the formula for the area of a circle

Substitute A = 49

Divide both sides by π

Take the square root of both sides

Evaluate, rounded to two decimal places

Example 4

A circle has an area of 47 mm2. What is its circumference, rounded to two decimal places?

Create a strategy

Use the formula A = πr 2 to find the radius so we can then find the circumference.

Apply the idea

Start with the formula for the area of a circle

Substitute the area

Divide both sides by π

Take the square root of both sides

Now we can substitute the radius into the circumference formula. mm

Start with the formula for the circumference

Substitute the radius

Evaluate, rounded to two decimal places

Idea summary

We can use the area formula to find the unknown radius of a circle if we know the area.

A is the area r is the radius of the circle

Practice questions

Choose your question path

Explorer

1ab, 2ab, 3ab, 4, 5a-c, 6ab, 7ab, 8ab, 9, 11–13, 16a, 17, 18, 20, 22

Understanding

1 Find the radius of a circle with diameter:

2 Find the diameter of a circle with radius:

Adventurer

1bd, 2bd, 3bd, 4, 5de, 6cd, 7cd, 8cd, 9, 10a, 12, 14, 16b, 18, 20–22

3 Calculate, rounding answers to two decimal places:

4 What fraction of a circle is shown in the diagram? a b r

Trailblazer

1cd, 2cd, 3cd, 5ef, 6cd, 7cd, 8cd, 9, 10b, 12, 15, 16d, 18–23

Fluency

5 Calculate the area of each circle, rounded to two decimal places:

6 Calculate the area of the circle with given dimension, rounded to one decimal place:

a Radius = 4 mm

c Radius = 12 mm

b Diameter = 10 mm

d Diameter = 0.44 m

7 Calculate the area of the circle with given dimension, rounded to two decimal places:

a Radius = 9 cm

c Radius = 12 mm

b Diameter = 24 cm

d Diameter = 13.2 m

8 For each circle with given information, round answers to two decimal places:

i Calculate the radius.

a Diameter = 12 mm

c Diameter = 22 mm

ii Calculate the area.

b Circumference = 18 cm

d Circumference = 43.98 cm

9 Find the area, rounded to one decimal place: a b c d

10 For each circle with given area:

i Calculate the radius, rounded to three decimal places.

ii Calculate the circumference, rounded to one decimal place.

a Area = 25 cm2 b Area = 121 cm2

11 A circle has an area of 144 mm2. Calculate the radius, rounded to three decimal places.

12 The area of a circle is 352 cm2

a If its radius is r cm, find r, rounded to two decimal places.

b Find the circumference of the circle, rounded to one decimal place.

13 A circle has an area of 144 cm2.

a Calculate the radius, rounded to two decimal places.

b Calculate the diameter, rounded to one decimal place.

14 A circle has an area of 81 mm2.

a Calculate the radius, rounded to two decimal places.

b Calculate the diameter, rounded to one decimal place.

15 A circle has an area of 169 mm2.

a Calculate the diameter, rounded to three decimal places.

b Calculate the circumference, rounded to one decimal place.

16 Find the radius of the circle with the given area, rounding your answer to one decimal place:

a Area = 64 cm2 b Area = 81 cm2 c Area = 36 cm2 d Area = 16 cm2

17 The radius of a circular baking tray is 10 cm. Find its area, rounded to two decimal places.

Reasoning

18 a Calculate the area of two circles, one with a radius of 4 cm and the other with a radius of 8 cm.

b By what factor is the area increased by when the radius is doubled?

c What happens to the area of a circle if we halve its diameter?

19 a What happens to the area of a circle if we triple its circumference?

b What happens to the area of a circle if we increase the radius by a factor of q?

c What happens to the radius of a circle if we increase the area by a factor of q?

Problem-solving

20 A pizza has a radius of 13 cm. Javier cuts himself a slice of pizza as shown in the diagram. What area of pizza, to the nearest whole number, is remaining?

21 The engineering team at Rocket Surgery are building a rocket for an upcoming Mars mission.

A critical piece is the circular connective disc that connects the booster rocket to the rest of the spacecraft. This disc must completely cover the top of the booster rocket.

The booster rocket has a diameter of precisely 713.5 centimetres. Round answers to two decimal places.

a Find the required area of the connective disc.

b Instead of using a calculator, an engineer uses the approximation 3.14 for π

Find the area using the engineer’s approximation for π.

c If the connecting disc is more than 100 cm2 too small, the disc will malfunction, resulting in catastrophic launch failure.

Will the disc malfunction if it is built according to the engineer’s calculation? Explain your answer.

22 Danielle deciding between buying a large pizza with a diameter of 46 centimetres for $18, or two medium pizzas with a diameter of 31 centimetres for $9 each. She thinks that buying two medium pizzas will give her more pizza for the same price. Is her decision justified? Explain your reasoning.

23 A wind turbine has blades that are R m long which are attached to a tower 60 m high.

When a blade is at its lowest point (pointing straight down), the distance between the tip of the blade and the ground is 20 m.

a Calculate the value of R.

b Find the distance travelled by the tip of the blade during one full revolution, rounded to two decimal places.

60 m R m

c A factor in the design of wind turbines is the amount of area covered by their blades. The larger the area covered, the more air can pass through the blades. Find the area inside the circle defined by the rotation of the blade tips, rounded to two decimal places.

10.05 Area of sectors

After this lesson, you will be able to…

• calculate the area of a sector given the contained angle and radius.

• solve mathematical and real-world problems involving the calculation of the area of sectors.

Area of a sector

Imagine you have a round pizza in front of you. When you take a slice, you are actually taking a ‘sector’ from the whole pizza.

A sector is the region inside a circle between two radii.

Similar to how the arc length is simply a fraction of the circumference, the area of a sector is simply a fraction of the circle’s area.

We can calculate the area of a sector by finding the area of the circle they are a part of and then taking the appropriate fraction. For example, the area of a semicircle is half the area of the full circle.

We can make a formula for the area of any sector depending on the contained angle θ that subtends the arc at the centre.

A sector with contained angle θ corresponds to a fraction of a full circle and so its area is given by:

The sector in the diagram has an angle of 30° and a radius of 6 cm.

a What fraction of the circle’s area is covered by this sector?

Create a strategy

To find the fraction make the angle of the sector the numerator and 360 the denominator.

Example 1

Apply the idea

Divide the angle of arc by 360

Simplify the fraction

b Find the exact area of the sector.

Create a strategy

Use the formula A = πr 2 and multiply the area by

Apply the idea

Multiply the area by

Substitute the radius

Evaluate the product

Simplify

Example 2

Find the area of the following sector of a circle. Round your answer to one decimal place.

Create a strategy

Use the formula for the area of sector:

Apply the idea

Start with the formula for the area of a sector

Substitute r = 13 and θ = 121

Evaluate and round

Idea summary

The area of the sector with contained angle θ can be found using:

θ is the contained angle of the sector r is the radius 13 cm 121°

Practice questions

Choose your question path

Explorer

1ef, 2–4, 8, 9, 13, 17

Understanding

Adventurer

1abef, 2–6, 8, 9, 11–13, 15–17

1 Find the fraction of the circle’s area covered by each sector:

Trailblazer

1acd, 4–7, 10–14, 16

2 Write the formula for the area of each sector:

3 For each circle:

i Find the fraction of the circle’s area covered by the sector.

ii Find the exact area of the sector.

4 Calculate the area of each sector, rounding your answers to one decimal place:

5 The area of the circle is 10 cm2. Find the area of the shaded sector. Round your answer to two decimal places.

6 Find the exact area of a semicircle with a radius of 15 cm.

7 Find the area of the sector of a circle of radius 16 cm if the sector subtends an angle of 78° at the centre. Round your answer to two decimal places.

8 In the diagram, O is the centre of a circle with radius 6 cm. Arc JK has a length measuring 2π cm.

Determine the exact area of sector OJK.

9 The given circle, centred at O, has radius 18 cm and arc AB has length 9 cm:

Find the area of the sector OAB

10 The area of the sector shown is 2960.92 m2.

a Find the length of the radius, rounded to one decimal place.

b Find the perimeter of the sector, rounded to one decimal place.

11 The diagram shows an arc JK of a circle, with centre O. The radius of the circle measures 15 cm and the arc measures 12 cm in length.

a Calculate ∠JOK to the nearest degree.

b Calculate the exact area of the sector.

Reasoning

12 A formula to find the perimeter of a sector when the angle θ is given in degrees is:

a If the perimeter of the following sector is 96.9 m, find the size of the angle θ, to the nearest degree.

b Find the area of the sector. Round your answer to the nearest integer.

c If we did not have the information from part (a), could we still solve part (b)? Justify your answer.

13 A large 17 m long sprinkler is placed in a crop field, with one end fixed and the other end free to move. As it rotates, it waters everything underneath it.

a If the sprinkler has rotated 86° since the farmer left, find the area of the crop field it has watered. Round your answer to one decimal place.

b Explain how the rotation angle of the sprinkler affects the area of the crop field it waters.

14 A security light shines on an adjacent fence at a distance of 10 metres from the light bulb.

a Find the area on which this security light shines.

b The light bulb is replaced with a new one that shines twice as bright. Explain how this change would affect the area on which the security light shines, assuming the distance to the fence remains the same.

Problem-solving

15 Roald makes a lady beetle cake with a diameter of 24 cm and a height of 10 cm. Roald removes a slice of the cake with an angle measure of 45°.

a Find the area that the black icing covers on the removed slice (both the top and side). Round your answer to one decimal place.

b Find the area that the red icing covers on the remaining cake (both the top and side). Round your answer to one decimal place.

16 The diagram shows a piece of jewellery made out of gold:

a Find the area of the piece. Round your answer to the nearest whole number.

b If the gold costs $4 per square millimetre, find the cost of the piece of jewellery.

17 The diagram shows the sectors of two concentric circles with common centre O where ∠O = 45°, OR = 8 cm, and OQ = 12 cm:

Calculate the area of the shaded region between the two circular arcs, rounded to one decimal place.

10.06 Area of composite shapes

After this lesson, you will be able to…

• find the area of composite shapes made up of rectangles, triangles, circles and other polygons using addition and subtraction methods.

Area of composite shapes

A composite shape is one that is made from a number of smaller shapes.

This shape is composed of a rectangle, square, triangle and parallelogram. 6

Sometimes shapes can be combined. This shape consists of a square with side lengths of 6 cm and two semicircles, which is equal to one full circle.

Area = Area of square + Area of circle

To calculate the area of a composite shape, we can use either of two methods:

• Addition method — Divide the composite shape into basic shapes, work out the area of each basic shape, then add them together.

• Subtraction method — Work out the area of the larger shape that encloses the composite shape, then subtract the areas of smaller basic shapes as necessary.

Interactive exploration

Explore online to answer the questions

Use the interactive exploration in 10.06 to answer these questions.

1. Identify the basic shapes that make up the composite shape shown in the applet.

2. How does the total area of the composite shape compare to the sum of the areas of the individual shapes?

Example 2

Find the area of the composite shape rounded to two decimal places.

Create a strategy

To find the area of this shape, subtract the area of the semicircle from the area of the trapezium.

Apply the idea

Start with the formula for the area of a trapezium

Substitute the values

Evaluate

Use the formula

Substitute the radius

Simplify

Subtract the areas

Evaluate and round the answer

Idea summary

To calculate the area of a composite shape, we can use either of two methods:

• Addition method — Divide the composite shape into basic shapes, work out the area of each basic shape, then add them together.

• Subtraction method — Work out the area of the basic shape that encloses the composite shape, then subtract the areas of smaller basic shapes as necessary.

Practice questions

Choose your question path

Explorer

Adventurer Trailblazer 1, 4a, 5i-a, 7ab, 9, 10a, 12, 13a 2, 3, 4ab, 5b, 6abc, 7a-d,e, 8ab, 9, 10ab, 11–13

2, 3, 4b, 5b, 6d, 7de, 8bcd, 10, 11, 13, 14

Understanding

1 Complete the statement to describe how to find the shaded area: a b

Total area = Area of ⬚ + Area of ⬚

Total area = Area of ⬚  Area of ⬚

2 Which method, addition of areas or subtraction of areas, would be the most efficient method to calculate the shaded area?

3 a Name the two shapes that make up the shaded area. b Which method, subtraction or addition, should be used to calculate the area?

Fluency

4 For each of these composite shapes: i What basic shapes make up the composite shape? ii Find the area, rounded to two decimal places.

5 For each of these composite shapes:

i What basic shapes make up the composite shape?

ii Find the area, rounded to two decimal places.

6 Find the area of the composite shapes. Round your answer to two decimal places.

7 Find the shaded area in the diagrams. Round your answer to one decimal place.

8 Find the shaded area, rounded to two decimal places:

Reasoning

9 Determine if each statement is true or false. Justify your response.

a The subtraction method is always faster than the addition method for finding the area of a composite shape.

b The area of a composite shape can be found by dividing it into multiple basic shapes and using the addition method.

c The area of a composite shape is always equal to the sum of the areas of its component shapes.

d If two composite shapes have the same perimeter, they will have the same area.

e If you know the area of a composite shape and the area of one of the shapes that makes it up, you can find the area of the remaining shapes.

10 The shape between two concentric circles as shown in the diagram is called an ‘annulus’.

a Calculate the area of the annulus rounded to one decimal place.

b If the radius of the large circle is R and the radius of the small circle is r write a formula to find the area of an annulus.

c Write your rule from part (b) in factorised form.

Problem-solving

11 Tricia is drawing a Yin Yang symbol. She chooses to make the radius of the overall circle 16 cm and the radius of the small circles 2 cm: Find the area of the shaded part, rounded to two decimal places.

12 The front door shown is 1 m wide and 3 m high and has four identical glass panels, each 76 cm long and 12 cm wide.

a Calculate the total area of the glass panels.

b The door is to be painted inside and outside. Calculate the total area to be painted.

13 The leadlight panel shown depicts a sunrise over the mountains. The mountain is represented by a green triangle 40 cm high. The yellow sun is represented by a sector of a circle with an 20 cm radius. There are 10 yellow sunrays in the shape of isosceles triangles with a base of 4 cm and a height of 12 cm. The sky is blue. If the mountain covers of the sun, calculate the area of the leadlight panel made of, rounded to two decimal places:

a green glass

b yellow glass

c blue glass

14 On each of the four sides of a square an equilateral triangle is drawn, the final figure resembling a four pointed star. If the sides of the square are cm long and the formula for the area of the equilateral triangle is , what is the area of the complete figure?

Chapter 10 review

1 Estimate the area of the butterfly if one unit on each axis represents 1.7 m.

2 A farm covers an area measuring 340 acres. Given that 1 acre is approximately 0.4 ha, and 1 ha is 10 000 m2, calculate:

a The area of the farm in hectares b The area of the farm in square metres

3 Calculate the area of the rectangles:

a length = 7 cm, width = 4 cm

c length = 15 m, width = 6 m

b length = 11 mm, width = 10 mm

d length 8 cm, width = 4.4 cm

4 A rectangular garden measures 15 metres by 10 metres.

a What is the area of the garden in square metres?

b If the walkway surrounding the garden is 3 metres wide, what is the total area of the garden and walkway in square metres?

5 Calculate the area of the parallelograms:

a base = 5 mm, height = 2 mm

b base = 10 m, height = 5 m

c base = 14 m, height = 5 m d base = 15 cm, height = 7 cm

6 A rectangular kitchen has a floor that measures 6 m by 5 m. The floor is to be covered with tiles that are a 30 cm square. How many tiles are required to completely cover the floor?

7 An area measuring 2280 cm2 is to be paved with identical tiles in the shape of parallelograms. Each tile measures 12 cm along the base, and has a perpendicular height of 5 cm.

a Find the area that each tile covers.

b How many tiles are needed to cover the whole area?

8 Find the area of each quadrilateral:

9 The given rhombus, with diagonal lengths x and y, is formed into a rectangle:

a Find the length of the rectangle in terms of x and/or y.

b Find the height of the rectangle in terms of x and/or y

c Find the formula for the area of the rhombus.

10 Find the value of k:

a A = 22.5 cm2 b A = 137.5 cm2

11 The following kite has an area of 48 cm2. The length of one of its diagonals is 12 cm:

Find the length of the other diagonal, k

12 For each of the following trapezia, find the value of the pronumeral:

a A = 42 mm2

b A = 20 mm2

13 For the rhombus in the diagram, one of the diagonals is of length 9 cm. How long is the other diagonal?

14 A gardener wants to plant a flower bed in The shape of a trapezium. The shorter base of the trapezium is 60 m and the longer base is 100 m. The height of the trapezium is 40 m. If the gardener can plant 2000 seeds per square metre, how many seeds will he need to plant the entire flower bed?

15 A company that makes kites produces them with diagonals of length 50 cm and 30 cm. The kites need to be coated with a resin that costs $2.30 per m². How much does it cost them to apply the coating to 1000 kites?

16 A landscaper charges $3 per square metre to maintain gardens. Answer these questions:

a How much would he charge to maintain a rectangular garden with dimensions 10 m by 7 m?

b How much would he charge to maintain a circular garden with a radius of 4 m?

c He charged $450 to maintain a garden in the shape of a square. What was the length of the side of the square garden?

17 Calculate the area of each circle, rounded to two decimal places.

a A circle with a radius of 3 m

c A circle with a radius of 7 mm

b A circle has a radius of 9 cm

d A circle with a diameter of 1.2 m

18 Find the radius of each circle, giving your answers to two decimal places:

a A circle with area of 54 cm2

c A circle with area of 42 cm2

b A circle with area of 36 cm2

d A circle with area of 22 cm2

19 For each circle with given information, round answers to two decimal places:

i Calculate the radius.

a Diameter = 12 mm

c Diameter = 22 mm

ii Calculate the area.

b Circumference = 18 cm

d Circumference = 43.98 cm

20 A pizza has a radius of 13 cm. Javier cuts himself a slice of pizza as shown in the diagram. What area of pizza, to the nearest whole number, is remaining?

21 Find the area of the following sectors. Round your answers to one decimal place.

22 A cicle has an area of 81 mm2.

a Calculate the exact diameter.

b Calculate the diameter rounded to two decimal places.

23 Richard is considering between buying a giant pizza with a diameter of 60 cm for $22, or two small pizzas with a diameter of 40 cm for $11 each. He assumes that buying two small pizzas will yield more pizza for an equivalent price. Is Richard’s assumption correct? Provide an explanation for your answer.

24 Find the exact area of a semicircle with a radius of 17 m.

25 Find the area of the sector of a circle of radius 17 cm if the sector subtends an angle of 89° at the centre. Round your answer to two decimal places.

26 Rashmir makes a lady beetle cake with a diameter of 24 cm and a height of 10 cm. Rashmir removes a slice of the cake with an angle measure of 45°.

a Find the area that the black icing covers on the removed slice (both the top and side). Round your answer to one decimal place.

b Find the area that the red icing covers on the remaining cake (both the top and side). Round your answer to one decimal place.

27 The diagram shows the sectors of two concentric circles with common centre O where radians, OR = 6 cm, and OQ = 10 cm.

Calculate the area of the shaded region between the two circular arcs, give your answer in terms of π

28 Find the area of each composite shape:

29 Find the area of each figure, rounded to one decimal place: