Unit 9 Section 2 : Area of Special Shapes

In this section we calculate the area of various shapes.

Area of a circle = π r²
Area of a triangle = b h
Area of a parallelogram = b h

Example 1

Calculate the area of the triangle shown.

Area  =  × 4 × 6  =  12 cm²

Example 2

Calculate the area of a circle with diameter 10 m.

Radius  =  10 ÷ 2  =  5 m

Area  =  π × 5²  =  78.53981634 m²  =  78.5 m² (to 3 significant figures)

Example 3

Calculate the area of the shape shown:

Area of rectangle  =  4 × 8  =  32 m²

Radius of semicircle  =  4 ÷ 2  =  2 m

Area of semicircle  =  × π × 2²  =  6.283185307 m²

Total area  =  32 + 6.283185307  =  38.283185307 m²  ≈  38.3 m²

Example 4

The diagram shows a piece of card in the shape of a parallelogram, that has had a circular hole cut in it.

Calculate the area of the shaded part.

Area of parallelogram  =  11 × 6  =  66 cm²

Radius of circle  =  4 ÷ 2  =  2 cm

Area of circle  =  π × 2²  =  12.56637061 cm²

Area of shape  =  66 – 12.56637061  =  53.43362939 cm²  ≈  53.4 cm²

Exercises

Question 1

Calculate the area of each of the following shapes:

(a)
m²
(b)
cm²
(c)
m²
(d)
cm²
Question 2

Calculate, giving your answers correct to 3 significant figures, the area of a circle with:

(a)
radius 6m, m²
(b)
diameter 20 cm, cm²
(c)
diameter 9 cm. cm²
Question 3

Calculate the area of each of the following shapes, giving your answers correct to 3 significant figures:

(a)
cm²
(b)
cm²
(c)
m²
(d)
cm²
Question 4

Calculate, giving your answers correct to 3 significant figures, the area of the semicircle with:

(a)
radius 30 cm, cm²
(b)
diameter 14 mm. mm²
Question 5

A circle of radius 8 cm is cut into 6 parts of equal size, as shown in the diagram.

Calculate the area of each part, giving your answer correct to 2 decimal places.

cm²
Question 6

Giving your answers correct to 3 significant figures, calculate the area of each of the following shapes. Each of the curved parts is a semicircle.

(a)
m²
(b)
cm²
(c)
mm²
(d)
cm²
Question 7

A rectangular metal plate is shown in the diagram. Four holes of diameter 8mm are drilled in the plate.

Calculate the area of the remaining metal, giving your answer correct to 2 decimal places.

mm²
Question 8

Calculate the area of the shape shown, giving your answer correct to 1 decimal place.

cm²
Question 9

The area that has been shaded in the diagram has an area of 21.8cm². Calculate the diameter of the semi-circular hole, giving your answer to the nearest millimetre.

mm
Question 10

The diagram shows the lid of a child's shape-sorter box. Calculate the area of the lid, giving your answer correct to 1 decimal place.

cm²
Question 11

Each shape in this question has an area of 10cm².
No diagram is drawn to scale.

(a)

Calculate the height of the parallelogram.

cm
(b)

Calculate the length of the base of the triangle.

cm
(c)

What might be the values of h, a and b in this trapezium?

What else might be the values of h, a and b ?

e.g. h:, a:, b:
Any set of values are correct for which a > b and (a + b) × h = 20
(d)

Look at this rectangle:

Calculate the value of x and use it to find the length and width of the rectangle.

Length = cm

Width = cm

2x + 2 = 10x - 1
4x + 3 = 10x
3 = 6x
x = 0.5
Length = 4x + 2 = 4 × 0.5 + 2 = 4cm
Width = = = 2.5cm
Question 12

This shape is designed using 3 semi-circles.

The radii of the semi-circles are 3a, 2a and a.

(a)

Find the area of each semi-circle, in terms of a and π, and show that the total area of the shape is 6πa².

Area first semicircle  =   π a²
Area second semicircle  =   π a²
Area third semicircle  =   π a²
Total area  =   π a²
(b)

The area, 6πa², of the shape is 12 cm² .

Calculate the area of a, giving your answer correct to 2 decimal place.

a = cm
6πa²  =  12 a²  =  a  = 
Question 13

Calculate the area of this triangle.

Use Pythagoras' theorem, see: Year 8 Unit 3

cm²

(Height of triangle)²  =  25² – 7²  =  625 – 49  =  576

Height of triangle  =  576  =  24 cm

Area of triangle  =  7 × 24 ÷ 2  =  84 cm²

Question 14

A box for coffee is in the shape of a hexagonal prism.

One end of the box is shown below.

Each of the 6 triangles in the hexagon has the same dimensions.

(a)

Calculate the total area of the hexagon.

cm²

Area of one triangle  =  5 × 4 ÷ 2  =  10 cm²

Area of hexagon  =  6 × 10  =  60 cm²

(b)

The box is 10cm long.

After packing, the coffee fills 80% of the box.

How many grams of coffee are in the box?

(The mass of 1 cm³ of coffee is 0.5 grams.)

grams

Volume of box  =  area of cross – section × length  =  60 × 10  =  600 cm³

Volume of coffee  =  80% of 600 cm³  =  0.8 × 600  =  480 cm³

Mass of coffee  =  0.5 × 480;  =  240 grams

(c)

A 227 g packet of the same coffee costs £2.19.
How much per 100 g of coffee is this?

£
The cost of 100 grams  =  £2.19 ÷ 227 × 100
 =  0.964757709  =  £0.96 (to the nearest penny)