M3.6.6 Chapter 6

Circles

How far is it all the way round a round table?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A About twice the distance across.
You picked A. Read on. At the end, the chapter comes back to your guess.
B About four times the distance across.
You picked B. Read on. At the end, the chapter comes back to your guess.
C A little more than three times the distance across.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Name the radius, diameter and circumference of a circle
Measure round tins to see that circumference ÷ diameter is about 3.14
Find the circumference and area of a circle

A wheel, a coin and a plate are all circles. A circle has three parts that you will use again and again.

The radius is a straight line from the centre to the edge. The diameter is a straight line across, through the centre. The circumference is the distance all the way round the edge.

Draw it first

The parts of a circle

1 Draw a circle. Mark its centre with a dot.
2 The radius goes from the centre to the edge. Here it is 3 cm.
3 The diameter goes across, through the centre. It is two radii, 6 cm.
4 The circumference is the whole edge, the distance all the way round.

The diameter is always twice the radius. So a radius of 3 cm means a diameter of 6 cm. If you know the diameter, halve it to find the radius.

Now a big question. How does the circumference compare with the diameter? Imagine the edge of a circle cut open and laid out straight.

The circumference is three diameters and a little more.

This is true for every circle, big or small. The circumference divided by the diameter is always the same number. That number is called pi, written π.

Pi is about 3.14. We use 3.14 because it is a rounded value of π. The digits of π go on for ever, so no number written in digits is exactly π.

So the circumference is π × the diameter. With 3.14 for π, circumference ≈ 3.14 × diameter.

Worked example

A circle has a diameter of 10 cm. What is its circumference? Use 3.14 for π.

The circumference is π × the diameter.
The diameter is 10 cm.
3.14 × 10 = 31.4 cm.

Answer: 31.4 cm

The diameter is 10 cm. Using 3.14, the circumference is about 31.4 cm.

Worked example

A circle has a radius of 7 m. What is its circumference? Use 3.14 for π.

The radius is 7 m. The diameter is double that: 2 × 7 = 14 m.
The circumference is π × the diameter.
3.14 × 14 = 43.96 m.

Answer: 43.96 m

The radius is 7 m. The diameter is 14 m. The circumference is about 43.96 m, from 3.14 × 14.

Now the area, the space inside a circle. Put the circle inside a square. The square has sides of two radii, so it is made of four small squares. Each small square is r by r.

The circle fills a little less than the four r × r squares.

The circle covers a little more than three of the four small squares. In fact it covers π of them, which is about 3.14. So the area of a circle is π × r × r. We write r × r as r².

Work out r × r first. Then multiply by 3.14. Use the radius, not the diameter.

Worked example

A circle has a radius of 5 cm. What is its area? Use 3.14 for π.

The area is π × r × r.
The radius is 5 cm, so r × r = 5 × 5 = 25 cm².
3.14 × 25 = 78.5 cm².
Check: the circle is smaller than the square round it. That square is 10 cm by 10 cm, which is 100 cm². And 78.5 is less than 100.

Answer: 78.5 cm²

Each small square is 5 × 5 = 25 cm². The circle is 3.14 of them: 78.5 cm².

See it three ways

With things

Wrap a string once round a tin. Straighten the string and measure it. Measure across the tin. Divide the first length by the second.

As a picture

In symbols

Circumference ÷ diameter ≈ 3.14. So circumference ≈ 3.14 × diameter, and area ≈ 3.14 × r × r.

Try it

Measure round tins

You need: three round tins or cups of different sizes, a piece of string, a ruler, a pencil.

Wrap the string once round the middle of a tin. Pinch the string where it meets itself.
Straighten the string and measure it with the ruler. This is the circumference.
Measure across the top of the tin, through the middle. This is the diameter.
Divide the circumference by the diameter. Do this for each tin and compare.

What you will see: Every answer is close to 3.1 or 3.2. It is not exactly 3.14, because string stretches and a ruler is read by eye. The answers are close to each other because the number is the same for every circle.

If you cannot try it, this is what happens.

A wrong idea many people have

To square the radius, you double it.

Why it feels right: For a radius of 2, both r × r and 2 × r give 4, so they look the same.

The test that breaks it: Try a radius of 5 cm. Then r × r is 5 × 5 = 25, but 2 × r is 10. The circle holds about 78.5 cm², not 3.14 × 10 = 31.4 cm².

What is true: r² means r × r. Multiply the radius by itself, then multiply by π.

Key words

radius: the distance from the centre of a circle to its edge.
diameter: the distance across a circle, through its centre. It is twice the radius.
circumference: the distance all the way round a circle.
pi: the number you get from circumference ÷ diameter. It is about 3.14.

Back to your guess

The answer is C: A little more than three times the distance across.

Go round any circle with a string, then lay the string along a line across the circle. The string is a little more than three times as long. The number is a little more than 3.14, and it is the same for every circle.

Keep it

Copy this into your exercise book.

Diameter = 2 × radius. Circumference = π × diameter, about 3.14 × diameter. Area = π × r × r. π is about 3.14, a rounded value.

Draw: A circle with its centre, a radius and a diameter drawn and named. Mark the edge thick and write circumference.