Maths

Circles

Circles are part of everyday life. Wheels, plates, CDs and coins are all examples of circles. Engineering and design tasks involving circles use a knowledge of area and circumference.

Words associated with circles

Here is a list of words associated with circles. How many do you know?

Put your mouse on each word to highlight an example.

You need to be familiar with the following abbreviations:

  • C = circumference
  • A = area
  • d = diameter
  • r = radius

Circumference of a circle

image: circumference of the circle

The circumference is the length of the edge around a circle.

For any circle, the circumference is:

3.141592... × the diameter.

Or in symbols: C = (3.141592...)d

This is true for all circles and so 3.141592... is therefore a special, unique number, and we represent it with the Greek letter π.(The symbol π is called 'pi' in English and is pronounced 'pie').

So we can write the formula for the circumference of a circle as: C = πd

However, the diameter is equal to 2 × radius, (2r), so we can also write this formula as:

C = 2πr

It does not matter which of these formulae you use. But you must be careful to use the correct length for the formula (the radius or diameter).

Look at the following example:

Question
image: circle split into 8, diameter: 50cm

Find the circumference of a wheel with a diameter of 50 cm. Use π = 3.14.

toggle answer

Answer

Solution 1: Using C = πd

We know that the diameter is 50 cm, so we simply calculate:

πd = 3.14 × 50 = 157 cm

Solution 2: Using C = 2πr

We have been given the diameter (50 cm), so we must start by finding the radius. The radius is half the diameter, so the radius is 25 cm.

2πr = 2 × 3.14 × 25 = 157 cm

Question

A plate has a radius of 15 cm. Find the circumference by using the π button on your calculator, and give your answer correct to 1 decimal place (dp).

image: circle, radius:15cm

toggle answer

Answer

The answer is 94.2 cm.

Solution 1 : Using C = πd

We must first calculate the diameter:

d = 2r, so d = 2 × 15 = 30 cm

C = π × 30 = 94.2477...

C = 94.2 cm (1 d.p.)

Solution 2 : Using C = 2πr

C = 2 × π × 15 = 94.2477...

C = 94.2 cm (1 dp)

Area of a circle

There is only one formula for the area of a circle:

A = π r2

We must therefore remember to use the radius each time.

Question

Find the area of a circle with a diameter of 20 cm. Use the π button on your calculator, and give the answer correct to 1 dp.

image: circle, diameter 20cm

toggle answer

Answer

We have been told that the diameter is 20 cm. Therefore, the radius is 10 cm.

A = π r2

A = π × 10 × 10 = 314.159... cm2

A = 314.2 cm2 (1 dp)

Remember

  • Units of area are always written as squares. For example, cm2, m2 etc.
  • πr2 means π × r × r (only the r is squared).

Now try this last question.

Question

Find the area of a circle with a radius of 12 cm.

Use the π button on your calculator, and give your answer correct to 3 significant figures (s.f.).

image: cirle radius 12cm

toggle answer

Answer

The answer is 452 cm2.

You need to calculate π × 12 × 12 = 452.3893... and then round your answer to 3 s.f.

Semi-circles and quarter circles

The perimeter of a semi-circle can be found by using the equation π x r², which is commonly used to calculate the area of a whole circle:

  • area of a semi-circle: 1/2 x π x r2
  • area of a quarter circle: 1/4 x π x r2
  • perimeter of a semi-circle: π x r + 2 x r
  • perimeter of a quarter circle: 1/2 x π x r + 2 x r

Example

A window in the shape of a semi-circle has a radius of 40 cm. Work out:

  1. The area
  2. The length of the sealant strip needed for the perimeter

Solution

    • Area = 1/2 x π x r2

    • 1/2 x π x 402

    • Area = 2513 cm2

    •  
    • Perimeter = π x r + 2 x r

    • π x 40 + 2 x 40

    • 125.7 + 80

    • Perimeter = 205.7 cm

    •  

Question

Jenny has a patio in her garden. Its shape is a quarter circle with a radius of 3m.

  1. How much space is there on the patio?
  2. What is its perimeter?

toggle answer

Answer
    • We need to find the area.

    • Area = 1/4 x π x r2

    • 1/4 x π x 32

    • Area = 7.07m2

    •  
    • Perimeter = 1/2 x π x r + 2 x r

    • 1/2 x π x 3 + 2 x 3

    • 4.7 + 6

    • Perimeter = 10.7m

Back to Revision Bite