circumference

If you wrap a string snugly around a tin can and then straighten the string out, its length is the circumference — the distance once around a circle, the circle's version of perimeter. The remarkable thing the ancients discovered is that this distance is always the same multiple of the diameter, no matter how big or small the circle.

That fixed multiple is pi (about 3.14159...), defined as the ratio of any circle's circumference to its diameter: pi = C / d. Rearranged, C = pi · d, and since the diameter is twice the radius, C = 2 · pi · r. A circle of radius 5 has circumference 2 · pi · 5 = 10 · pi ≈ 31.4. Part of the way around — an arc cut off by a central angle theta in radians — has length r · theta, or for an angle in degrees, (theta / 360) · 2 · pi · r.

Pi is not a fraction in disguise: it is irrational, so its decimal never ends and never repeats, and in fact it is transcendental (it is not the root of any polynomial with whole-number coefficients). The fractions 22/7 and 3.14 are handy approximations, not the true value — a fact with a famous consequence, since the transcendence of pi is exactly why squaring the circle with compass and straightedge is impossible.