Measuring a Circle

The distance around a circle is its circumference. The distance across, through the centre, is its diameter.

Measure both and divide

For every circle ever measured, circumference divided by diameter comes to about 3.14.

That fixed ratio has a name.

Pi is the circumference of any circle divided by its diameter, about 3.14.

It is a constant of proportionality

C = πd is y = kx with k = π. Circumference is proportional to diameter.

Where π actually comes from

Measure any circle's circumference and divide by its diameter. The answer is always about 3.14, whatever the circle. That constant ratio is π. It was discovered by measurement, not invented, and it is the same for every circle in the universe.

The two forms of the formula

Circumference = π × diameter, or equivalently 2 × π × radius, since the diameter is twice the radius. Both say the same thing. Check which measurement a problem gives you before choosing a form.

π never terminates or repeats

π is irrational: its decimal goes on forever without a repeating pattern. 3.14 and 22/7 are approximations, close enough for most purposes but never exact. Leaving an answer as "12π" is more accurate than any decimal.

Measure it yourself

Wrap a string round a tin, measure the string and the diameter, and divide. You will get something near 3.1. Doing this once with real objects converts π from a number in a formula into a fact about the world.