Discovering Pi

How many diameters wrap a circle?

In the last lesson, circumference meant the distance around a circle and diameter meant the distance across the circle through the centre.

Here's the question this whole lesson hangs on. Take any circle — a hockey puck, a dinner plate, a wagon wheel at Heritage Park. Cut a string exactly one diameter long. How many of those strings does it take to wrap all the way around the rim? Two? Four? And the sneakier question underneath: does a bigger circle need more of them?

Lock in a guess, then unroll for real. (Real string and a tape measure work too — the on-screen version just never slips.)

The number with a name

The circle changed. The diameter changed. The circumference changed. But circumference divided by diameter refused to move:

A ratio that loyal earns a name. We call it pi and write it with the Greek letter:

Pi is not a new measurement. It is the ratio that connects every circle's circumference to its diameter — the number your unroll table kept landing on.

Turn the ratio into a formula

If

then the circumference is about 3.14 times the diameter:

Using pi, the formula is:

In this course, use $\pi \approx 3.14$ unless a question tells you otherwise.

You can also run the formula backwards. If you know the circumference, divide by pi to find the diameter:

For example:

A circle with circumference 18.84 cm has diameter 6 cm.

Where it shows up in real life

A bike wheel moves forward about one circumference every time it makes one full rotation.

If a kid's bike wheel has diameter 50 cm, its circumference is:

So one full rotation moves the bike about 157 cm. After 100 rotations, the bike has moved about 15,700 cm, or 157 m.

Misconception probe

Worksheet

These are not graded. The goal is to connect the measurement pattern to the formula and to practise both directions.

Going further

Circumference measures the outside edge of the circle. The next circle lesson measures the inside: area. Same circle, different kind of size.