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.

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

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 = **π**d**

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

**C = 2**π**r**

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

- Answer
**Solution 1**: Using C = πdWe know that the diameter is 50 cm, so we simply calculate:

πd = 3.14 × 50 = 157 cm

**Solution 2**: Using C = 2πrWe 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).

- Answer
The answer is 94.2 cm.

**Solution 1**: Using C = πdWe 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 = 2πrC = 2 × π × 15 = 94.2477...

C =

**94.2 cm (1 dp)**

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

A = π r^{2}

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.

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

A = π r

^{2}A = π × 10 × 10 = 314.159... cm

^{2}A = 314.2 cm

^{2}(1 dp)

**Remember**

- Units of area are always written as squares. For example, cm
^{2}, m^{2}etc. - πr
^{2}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.).

- Answer
The answer is 452 cm

^{2}.You need to calculate π × 12 × 12 = 452.3893... and then round your answer to 3 s.f.

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 r^{2} - area of a quarter circle:
^{1}/_{4}x π x r^{2} - 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:

- The area
- The length of the sealant strip needed for the perimeter

**Solution**

Area =

^{1}/_{2}x π x r^{2}^{1}/_{2}x π x 40^{2}Area = 2513 cm

^{2}

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.

- How much space is there on the patio?
- What is its perimeter?

- Answer
We need to find the area.

Area =

^{1}/_{4}x π x r^{2}^{1}/_{4}x π x 3^{2}Area = 7.07m

^{2}

Perimeter =

^{1}/_{2}x π x r + 2 x r^{1}/_{2}x π x 3 + 2 x 34.7 + 6

Perimeter = 10.7m

Now try a Test Bite