Circumference of a Circle and the Meaning of Pi

1. The parts of a circle

A circle is the set of all points that are the same distance from one fixed point, called the centre. Every circular object you know, such as the base of a tin, a bowl, a car tyre or a roundabout, has the same parts.

▶ Name each part as it appears. Remember: diameter = 2 × radius.

• The circumference is the boundary: the distance round the circle.
• The radius joins the centre to the circumference. All radii of one circle are equal.
• The diameter is a chord (a straight line joining two points on the circle) that passes through the centre. It is the longest chord and equals two radii, so $d=2r$ and $r=\frac{d}{2}$.
• An arc is a part of the circumference. A sector is the part of the circle between two radii and the arc between them, like a slice of pizza or a piece of watermelon.

2. Measuring and discovering pi

To measure the diameter of a tin, place a ruler across its widest part, through the centre of the top. To measure the circumference you cannot use a straight ruler. Instead you wrap a string or a tailor's tape measure once round the tin, mark where it meets, straighten it and read the length on a ruler. For big circles such as a roundabout or a tyre you can use a measuring wheel or a long tape.

Now measure several round objects and, for each one, divide the circumference $c$ by the diameter $d$. Here are some results a class might record:

Whether the circle is small or large, the answer to $c÷d$ is a little bit more than 3. The ratio of the circumference to the diameter is the same number for every circle. Mathematicians gave it a name, the Greek letter pi, written $\pi$. It is not a whole number: its decimal digits never end (3.14159265...). So we use approximations:

• $\pi \approx \frac{22}{7}$ — a good choice when the radius or diameter is a multiple of 7.
• $\pi \approx 3.14$ — a good choice for other numbers. Some questions use 3.142.
• Always use the value of $\pi$ that the question tells you to use.

▶ Unroll the circumference: it fits the diameter just over three times. That 'just over 3' is pi.

3. The formula for the circumference

From the experiment we learned that $\frac{c}{d}=\pi$ for every circle. Multiplying both sides by $d$ gives the formula for the circumference: $$C=\pi d$$ In words: circumference = pi × diameter.

Since the diameter is twice the radius ($d=2r$), we can also write $$C=2\pi r$$ The two formulas give exactly the same answer. Use $C=\pi d$ when you are given the diameter and $C=2\pi r$ when you are given the radius.

1. 1Read the question and note whether you are given the radius or the diameter, and which value of $\pi$ to use.
2. 2Write the formula and substitute, e.g. $C=2\times \frac{22}{7}\times 140$.
3. 3Calculate carefully. With $\frac{22}{7}$, divide the number by 7 first when you can.
4. 4Write the answer with its unit (cm, m, km) and round if asked.

The formula also works backwards. If the circumference is known, the diameter is $d=\frac{C}{\pi }$ and the radius is $r=\frac{C}{2\pi }$. For instance, a circular table has circumference 176 cm. With $\pi =\frac{22}{7}$: $d=176÷\frac{22}{7}=176\times \frac{7}{22}=56$ cm.

4. Using the circumference in real life

The circumference helps us with everything that is round or rolls. The distance a wheel travels in one complete turn equals its circumference. So the distance for many turns is $\text{distance}=\text{number of turns}\times C$. A bicycle wheel of diameter 70 cm has $C=\frac{22}{7}\times 70=220$ cm, so in 500 turns it travels $500\times 220=110,000$ cm, which is 1,100 m.

To fence a circular pond, to put a border round a round table, or to tie a rope round a baobab tree, you need the circumference. If a question asks you to round, do so only at the end. For example, with $\pi =3.142$ and $r=6$ cm, $C=2\times 3.142\times 6=37.704$ cm, which is 37.7 cm to the nearest tenth.

Worked examples

Try each one yourself first, then open the solution and compare every step.