Year 7AC9M7M03

Circles — Circumference and Area

Describe the relationship between π and the features of circles including the circumference, radius and diameter. Note: the <b>area</b> of a circle is Year 8 (AC9M8M03), so that part is extension.

Two formulas. One uses the diameter, the other uses the radius squared. Mixing them up is the only real trap.

Method

Radius and diameter

d = 2r

The diameter is twice the radius.

Circumference

C = \pi d = 2\pi r

The distance all the way round.

Area

A = \pi r^{2}

Always the radius, always squared.

\pi \approx 3.14

Use the $\pi$ key on your calculator unless told otherwise.

Worked example

e.g.

\text{a circle with radius } 7 \text{ cm}

Write down what you are given $r = 7 \text{ cm}$
For circumference, use $C = 2\pi r$ $C = 2 \times \pi \times 7$
Work it out $C = 43.98 \text{ cm}$
For area, use $A = \pi r^{2}$ . Square the radius FIRST, then multiply by $\pi$ $A = \pi \times 7^{2} = \pi \times 49$
Work it out $A = 153.94 \text{ cm}^{2}$
Check the units. Circumference is a length, area is squared $\text{cm and cm}^{2}$

Common mistakes

Wrong

A = \pi \times (2 \times 7)^{2}

Right

A = \pi \times 7^{2}

Area uses the RADIUS, not the diameter. If you are given the diameter, halve it first.

Wrong

A = (\pi \times 7)^{2}

Right

A = \pi \times 7^{2}

Square the radius first, then multiply by

\pi

. Only the

r

is squared.

Wrong

C = \pi r

Right

C = \pi d = 2\pi r

Circumference uses the whole diameter. Forgetting the 2 halves your answer.

Practice

Give answers to 2 decimal places:

\text{Circumference, } r = 5

Answer

31.42

\text{Area, } r = 5

Answer

78.54

\text{Circumference, } d = 10

Answer

31.42

\text{Area, } r = 3

Answer

28.27

\text{Circumference, } r = 12

Answer

75.40

\text{Area, } d = 8 \ (\text{so } r = 4)

Answer

50.27

\text{Area, } r = 10

Answer

314.16

\text{Circumference, } d = 21

Answer

65.97

\text{A circle has } C = 31.42. \text{ Find } r

Answer

5

\text{A circle has } A = 78.54. \text{ Find } r

Answer

5