Take any circle. Measure the distance around its edge — that's the circumference. Measure the distance straight across through the center — that's the diameter. Now divide the first number by the second. No matter how big or small the circle is, you get the same answer every time: approximately 3.14159. That number is pi, written $\pi$.

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

This is the actual definition of pi — not "3.14," which is just a rounded-off approximation, and not some mysterious constant that shows up in circle formulas. Pi is the ratio $C/d$. Everything else about pi — its appearance in area formulas, trigonometry, physics — flows from this one relationship.

A natural question: why should this ratio be the same for a bike tire and for the Death Star? The answer is similarity. Any two circles are scaled copies of each other — same shape, different size. If you double a circle's diameter, you also double its circumference, because you're stretching every part of the circle by the same factor. Since both circumference and diameter scale up by the same multiple, their ratio stays fixed. Compare this to a ratio like height/weight for people, which changes from person to person because height and weight don't scale together in a fixed way. Circles are special: stretching a circle uniformly is the only thing you can do to it, so the ratio of any two linear measurements on a circle is locked in.

You'll often see the radius used instead of the diameter. The radius is the distance from the center to the edge — exactly half the diameter, so $d=2r$. Substituting into the definition of pi gives the circumference formula most people memorize:

$$C=\pi d=2\pi r$$

A common mixup is grabbing the radius when a problem gives you the diameter, or vice versa. Always check which one you're handed before plugging into a formula.

Pi also governs the circle's interior. The area of a circle — the amount of surface it encloses — is:

$$A=\pi r^2$$