Fractions
Decimals
Length and area
Solids and volume
Probability
Big ideas
The formula for the area of a triangle is derived from the area of a rectangle, then applied to find the area of composite shapes and surface area.
Curriculum Outcomes
• Establish and apply relationships between lengths of sides, perimeter and area for squares, rectangles and triangles. Generalise and apply formulas, using appropriate units
• Explore and explain efficient strategies to determine the perimeter and area of irregular or composite shapes composed of squares and rectangles
• Explore and establish connections and conversions between units of area
• Analyse, represent and interpret problems in real-world situations involving perimeter and area of squares, rectangles, triangles and rectangular composite shapes, parallel lines, properties of triangles, transformations of points, views of rectangular prisms and rectangular composite objects, volume and/or Australian time zones
12 Length and area
Chapter outline
Area of a triangle
The area of a triangle is given by
Example 1
Find the area of this triangle:
Create a strategy
Use the formula for the area of a triangle.
Apply the idea
Write the formula
Substitute b = 10 and h = 7
Evaluate
Example 2
Find the area of the given triangle with a base length of 10 metres and a perpendicular height of 8 metres.
Create a strategy
Use the formula for the area of a triangle.
Apply the idea
For this triangle, the base is b = 10 m and the height is h = 8 m.
Write the formula
Substitute b and h
Evaluate
3 For each figure:
3a-c, 4, 5a-c, 6ac, 7, 8ab, 9a, 10
3b-d, 4, 5a-d, 6ac, 7, 8ac, 9ac, 10
3d-f, 4, 5c-f, 6bd, 7, 8cd, 9bd, 10
i Find the area of the entire rectangle. ii Find the area of the triangle.
4 Find the area of each triangle:
Area of triangles
5 Find the area of a triangle which has these dimensions:
a Height = 8 cm and base = 9 cm b Height = 4 m and base = 6 m
c Height = 7 mm and base = 8 mm d Height = 3 cm and base = 9 cm
e Base = 12 cm and height = 100 mm f Base = 5.2 m and height = 200 cm
6 Determine whether these dimensions could form a triangle with an area of 20 mm2:
a Base = 10 mm, height = 4 mm b Base = 1 mm, height = 20 mm
c Base = 5 mm, height = 8 mm d Base = 2 mm, height = 2 cm
7 For this triangle with an area of 20 mm2, find the value of b.
8 Find the base measurement for these triangles given the area and height:
a Area = 18 cm2 and height = 9 cm b Area = 40 cm2 and height = 10 cm
c Area = 12.5 mm2 and height = 10 mm d Area = 400 mm2 and height = 4 cm
9 Find the height measurement for these triangles given the area and base:
a Area = 90 cm2 and base = 20 cm
b Area = 150 cm2 and base = 50 cm
c Area = 33.75 mm2 and base = 4.5 mm d Area = 2500 mm2 and base = 50 cm
10 Complete this table of base and height measurements for four different triangles, which all have an area of 30 m2:
Reasoning 11, 14 11, 12, 14, 16
11 Explain how the area of a triangle is connected to the area of a rectangle using this diagram:
13, 15, 16
12 A right-angled triangle has a base of 5 cm and a height of 10 cm.
a If the height is halved, what effect does it have on the area of the triangle?
b If the base is doubled, what effect does it have on the area of the triangle?
13 Two right triangles have the same area. If one triangle has a base of 8 cm and a height of 6 cm, what are the possible dimensions of the other triangle, given the base and height are whole numbers?
14 If two right-angled triangles have the same area, does this mean they have exactly the same shape? Explain your reasoning.
15 A triangle has base k units and perpendicular height p units.
a Write an algebraic rule to find A, the area of the triangle.
b If p = 5 mm and k = 20 mm, calculate the area of the triangle.
16 The formula to find the area of a triangle
a Rearrange the formula to make b the subject (make it b = ).
b Rearrange the formula to make h the subject.
Problem-solving 17, 18, 20, 21 18, 19, 21 19, 21–24
17 A farmer wants to install a triangular shade sail in her paddock to protect her sheep from the sun. The base of the shade sail is 12 m long and the height of the triangle is 8 m. What is the area of the triangular shade sail in square metres?
18 A civil engineer needs to design a triangular traffic island for a new road intersection. The base of the triangle is 20 m long and the height of the triangle is 15 m. Calculate the area of the triangular traffic island in square metres.
19 An artist is painting a triangular canvas with a base of 80 cm and a height of 50 cm. Calculate the area of the canvas that needs to be painted: a in square centimetres b in square metres
20 An architect is designing a triangular courtyard for a new apartment building. The base of the triangle is 30 m long and the height of the triangle is 20 m.
a Calculate the area of the courtyard.
b If paving tiles cost $25 per square metre, calculate the cost of paving the courtyard.
21 A painter charges $30 per square metre to paint walls. If a room has a triangular section with a base of 8 m and a height of 5 m, how much will it cost to paint that section of the wall?
22 Joy wants to paint her living room walls. One of the walls measures 3 m high and 5 m wide, while the other wall measures 3 m high and 4 m wide. There is a triangular window on one of the walls that has base of 2 m and height of 1 m. If each litre of paint can cover an area of 6 m2, how many exact litres of paint will Joy need to buy?
23 Cody is re-tiling his rectangular bathroom floor. The bathroom measures 2 m by 3 m, but there is a shower cubicle in the corner in the shape of a right triangle with a base of 1 m and a height of 1.5 m. Each tile measures 20 cm by 20 cm.
a Calculate the number of tiles that Cody needs to buy.
b If Cody needs to account for 10% wastage, how many tiles does he need to buy?