Circle Calculator

Know one measurement of a circle and you know them all. Pick which one you are starting from — radius, diameter, circumference or area — type it in, and the other three appear, with the formulas d = 2r, C = 2πr and A = πr² shown beside the results so the relationships stay visible rather than being taken on trust.

The relationship worth internalising is that the four quantities do not scale together. Diameter and circumference are linear in the radius; area is quadratic. Double a circle's width and you double the distance around it but quadruple the space inside it, which is why a 16-inch pizza has about 78 percent more surface than a 12-inch one for what usually looks like a modest price difference. The same arithmetic runs the other way for anything where the area is the cost: a pipe twice the bore carries roughly four times the flow at a given speed, and a tank twice the diameter holds four times as much per unit of height.

π is why none of these results is ever exact. It is irrational, so it has no finite decimal and no fractional form at all — and beyond that it is transcendental, proved so by Lindemann in 1882, which is the result that finally settled the ancient question of squaring the circle: you cannot build a square of equal area with compass and straightedge, because such a construction can only produce solutions to polynomial equations and π is not one. The familiar 22/7 is about 0.04 percent high, which is plenty for sketching but visibly wrong over a long circumference; 355/113 is a far better shortcut, matching π to seven digits.

For anything you are about to cut, measure the diameter rather than deriving it. Diameter is the one dimension you can take directly across a physical object with a rule, where radius requires finding a centre you cannot see and circumference requires wrapping something that may not sit flat. And nominal sizes lie: a pipe or a dowel sold as a given size is usually not that size to the millimetre, so measure the actual object before trusting the label.