Circumference calculator

The circumference of a circle is C = πd, pi times the diameter, or equivalently C = 2πr, two pi times the radius. It is the distance around the circle, its perimeter. For a diameter of 14 the circumference is 14π, about 43.9823. With the radius of 7 it is 2 × π × 7, the same 14π.

That formula is not derived from something else; it is what π means. π is defined as the ratio of any circle's circumference to its diameter, and the remarkable fact is that the ratio is the same for every circle, about 3.14159. So the distance around is always a little more than three times the distance across. Wrap a string around a can and lay it next to the can's width, and it goes just over three times.

The well-known approximation 22/7 works neatly with this example: 22/7 × 14 = 44 exactly, against the true 43.9823, too high by about 0.04%. 3.14 gives 43.96, a little low. Both are fine for cutting a length of edging or trim; for an answer "in terms of π", write 14π.

The mistakes are mostly about which length goes in. Using the radius in C = πd gives 7π, about 21.99, half the right answer. Using the diameter in C = 2πr gives 28π, about 87.96, twice the right answer. The other mix-up is with area: πr² is the space inside, 49π square units here, while the circumference is a length in plain units (centimeters, not square centimeters).

Working backwards is often the more useful direction, because the circumference is the easier thing to measure. You cannot get a ruler across a tree trunk or a pipe, but you can wrap a tape around it. Divide by π for the diameter: a trunk 100 cm around is 100 / π, about 31.83 cm across. Divide by 2π for the radius. Choose circumference as the measurement above and the page does that, and gives the area too.

A wheel turns this into distance. Each revolution rolls the wheel forward by one circumference, so a wheel 14 inches across travels 14π, about 43.98 inches, per turn. A mile is 63,360 inches, so it turns about 1,440.6 times per mile. Bicycle computers and odometers count exactly this way, and a wrong tire size throws them off by the same proportion.

Circumference is the whole of the curve. A part of it is an arc, the same fraction of 2πr that its angle is of 360°, and this calculator's arc mode computes it. For the perimeter of an ellipse there is no simple formula at all, and the area calculator uses Ramanujan's approximation for that. Doubling the diameter doubles the circumference, but quadruples the area.