Pi

A measurement, not a decree

$\pi$ is defined by a division you could do with string:

$$\pi = \frac{C}{d}$$

Measure a circle’s circumference, measure its diameter, divide. The answer is about $3.14159$ every time.

Circle	$C$	$d$	$C \div d$
a coin	$7.54$ cm	$2.4$ cm	$3.14$
a wheel	$2.20$ m	$0.7$ m	$3.14$

Why it never changes

Every circle is the same shape at a different size. Scaling one up multiplies its circumference and its diameter by the same factor, so the ratio between them cannot move.

That constancy is what earns $\pi$ a name — it is the one number every circle has in common.

Where it appears

$$C = \pi d \qquad A = \pi r^2$$

The first is the definition rearranged. The second needs more work to justify, but the same $\pi$ is doing the job: converting a square into a circle.

It is irrational

$\pi$ cannot be written exactly as any fraction, and its decimal never repeats. $\tfrac{22}{7}$ and $3.14$ are approximations, close enough for schoolwork and not equal to it.

Leaving an answer as $12\pi$ is therefore more exact than writing $37.7$. Round only when a question asks for a decimal, and round once at the end — using $3$ instead of $\pi$ makes every answer about $5\%$ too small.