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Unit C3 Section 2
Area and Circumference of Circles

The circumference of a circle can be calculated using

C = 2πr or C = πd

where r is the radius and d the diameter of the circle.

The area of a circle is found using

A = πr2 or A =

Worked Examples

1

Find the circumference and area of this circle.

The circumference is found using C = 2πr , which in this case gives

C = 2π × 4
= 25.1 cm (to one decimal place)

The area is found using A = πr2, which gives

A = π × 42
= 50.3 cm² (to one decimal place)
2

Find the radius of a circle if:

(a)

its circumference is 32 cm,

Using C = 2πr gives
32 = 2πr
and dividing by 2π gives
= r
so that
r = 5.09 cm (to 2 decimal places)
(b)

its area is 14.3 cm².

Using A = πr2 gives
14.3 = πr2
Dividing by π gives
= r2
Then taking the square root of both sides gives
= r
so that
r = 2.13 cm (to 2 decimal places)
3

Find the area of the door shown in the diagram. The top part of the door is a semicircle.

First find the area of the rectangle.

Area = 80 × 160
= 12800 cm²

Then find the area of the semicircle.

Area = × π × 402
= 2513 cm²
Total area = 12800 + 2513
= 15313 cm² (to the nearest cm²)

Exercises

Find the circumference and area of each of the following circles.

(a)

circumference = cm (to 1 d.p.)

area = cm² (to 1 d.p.)
(b)

circumference = m (to 2 d.p.)

area = m² (to 2 d.p.)
(c)

circumference = m (to 2 d.p.)

area = m² (to 2 d.p.)
(d)

circumference = cm (to 1 d.p.)

area = cm² (to 2 d.p.)
(e)

circumference = m (to 1 d.p.)

area = m² (to 2 d.p.)
(f)

circumference = m (to 2 d.p.)

area = m² (to 2 d.p.)

Find the radius of the circle which has:

(a)
a circumference of 42 cm, r = cm (to 1 d.p.)
(b)
a circumference of 18 cm, r = cm (to 1 d.p.)
(c)
an area of 69.4 cm², r = cm (to 1 d.p.)
(d)
an area of 91.6 cm². r = cm (to 1 d.p.)

The diagram shows a running track.

(a)

Find the length of one complete circuit of the track.

m (to the nearest m)
(b)

Find the area enclosed by the track.

r = m² (to the nearest m²)

An egg, fried perfectly, can be thought of as a circle (the yolk) within a larger circle (the white).

(a)

Find the area of the smaller circle that represents the surface of the yolk.

cm² (to 2 d.p.)
(b)

Find the area of the surface of the whole egg.

cm² (to 2 d.p.)
(c)

Find the area of the surface of the white of the egg, to the nearest cm² .

cm² (to the nearest cm²)

The shapes shown below were cut out of card, ready to make cones.

Find the area of each shape.

(a)
cm² (to 2 d.p.)
(b)
cm² (to 1 d.p.)

A circular hole with diameter 5 cm is cut out of a rectangular metal plate of length 10 cm and width 7 cm. Find the area of the plate when the hole has been cut out.

cm² (to 2 d.p.)

Four semicircles are fixed to the sides of a square as shown in the diagram, to form a design for a table top.

(a)

Find the area of the table top if the square has sides of length 1.5 m.

m² (to 2 d.p.)
(b)

Find the length of the sides of the square and the total area of the table top if the area of each semicircle is 1 m².

length of the sides = m (to 1 d.p.)

total area = m² (to 2 d.p.)

The radius of a circle is 8 cm.

Work out the area of the circle.

(Use π = 3.14 or the π button on your calculator.)

cm² (to the nearest cm²)

The diameter of a garden roller is 0.4 m.

The roller is used on a path of length 20 m.

Calculate how many times the roller rotates when rolling the length of the path once.

Take π to be 3.14 or use the π key on your calculator.

times