Circles: Area and circumference

In a nutshell

All circles have a diameter and a radius. In combination with the Greek letter $$\pi$$, these properties can be used to calculate the area and perimeter of a circle. 

Diameter and radius

The diameter is a straight line from one side of a circle to another which passes through the centre. The radius is a straight line from the centre of a circle to its side. 

The radius and diameter are linked by the formula:

 $$d = 2r$$​​

​Where $$r$$​​ is the radius of the circle, and $$d$$​​ is the diameter.

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Example 1

What property of a circle is shown by the line $$OA$$?​

The line passes from the centre of a circle to a point on the side labelled $$A$$.

This defines the radius.

The line OA is the radius of the circle.

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Circumference of a circle

The circumference is the name given to the perimeter of the circle. The Greek letter $$\pi$$​ is a constant ratio between the circumference of a circle and its diameter.

The circumference of a circle is given by:

$$Circumference=2\pi r = \pi d$$​​

​$$\pi \approx 3.142$$​​

Note: The value of $$\pi$$ on a calculator can be used which will be more accurate. 

Example 2

What is the circumference of a circle which has a radius of $$3cm$$? Leave your answer in terms of $$\pi$$.

​

Use the radius to calculate the diameter. 

​$$2 \times 3 = 6cm$$​​

Multiply the diameter by $$\pi$$.​

​$$\underline{6 \times \pi = 6\pi \ cm}$$​​

Note: Remember to maintain the units throughout the question. 

Area of a circle

The area of  circle is the size of its surface. 

The area is given by the formula:

​$$\text{Area}=\pi r^2$$​​

Example 3

The circle below has a radius of $$5cm$$.  What is the area in terms of $$\pi$$?

Identify the formula for the area of a circle.

​$${Area}=\pi r^2$$​​

Use the formula to calculate the area. 

$${Area}=5^2 \times \pi$$​​

​$$\underline{{Area}=25\pi \ cm^2 }$$