## The circumference of a circle

The circumference of a circle is the distance around the circle. The ancients, probably by using string, discovered that doubling the diameter of a circle doubled the circle's circumference; trebling the diameter of a circle trebled the circle's circumference and so on. In general, the circumference of a circle is proportional to its diameter.

That is, the ratio $$\dfrac{\text{circumference}}{\text{diameter}}$$ is the same for all circles.

We represent this constant ratio by the Greek letter π (pronounced 'pie' but spelt ‘pi’). Thus in every circle,

$$\dfrac{\text{circumference}}{\text{diameter}}=\pi$$

Rearranging this formula we obtain

$$C = \pi d$$

where C is the circumference and d is the diameter of the circle. Since the diameter is twice the radius, we can also write

$$C = 2\pi r$$

where r is the radius of the circle.

The number π is not a whole number, nor is it a rational number. Its approximate value, correct to seven decimal places, is 3.1415927, but the decimal expansion of π continues forever with no apparent pattern. This number is one of the most remarkable of all numbers in mathematics and reappears somewhat mysteriously in many places.

We usually round it off to two decimal places as 3.14. If greater accuracy is required, we can take more decimal places.

Although π is not a rational number (this is not easy to prove), it is close to the rational number $$3\dfrac{1}{7} = \dfrac{22}{7}.$$ Written as a decimal $$3\dfrac{1}{7} = 3.\dot14285\dot7$$ so $$3\dfrac{1}{7}$$ differs from π only in its third decimal place. This approximation is often used in situations where we wish to avoid decimals.

When solving problems involving π, it is best to leave the number in terms of π until the end of the problem and then, if required, use an approximation.