Circumference Calculator

Background.

Wrap a rope snugly around the equator, about 40,075 km of it, then add a single metre and lift the loop evenly off the ground. It stands 15.9 cm clear all the way round, high enough for a cat to squeeze under, and it would stand exactly as high if the rope had been wrapped around a football instead. The reason is the simplest fixed ratio in geometry: circumference is π times the diameter, so every extra metre of boundary buys 1/π of a metre of diameter, or 1/(2π), about 0.159 m, of radius, whatever the size of the circle.

That ratio, 3.14159…, has been chased for close to four thousand years. The Rhind Papyrus, an Egyptian scribe's copy from around 1650 BCE, implies a value near 3.16. Archimedes, in the third century BCE, trapped the circle between inscribed and circumscribed polygons of 96 sides and proved that π lies between 223/71 and 22/7. Around 480 CE the Chinese mathematician Zu Chongzhi pinned it to seven decimal places. The value carried here, to 15 decimals, is more precise than any tape measure will ever need.

Most everyday uses are about wrapping and rolling. A wheel moves forward one circumference per turn, which is how odometers and bicycle computers count distance. A seal around a hatch, a band around a tank, the steel plate rolled into a pipe and the belt over a pulley are all circumference problems. So is the reverse: foresters tape a tree's girth and divide by π to get its diameter.

Enter a radius or a diameter in any unit and the circumference comes back in that same unit, with the diameter shown beside it.

What is circumference calculator?

Circumference is the length of a circle's boundary — the distance you would cover walking once around it. It is a length, in the same unit as the radius or diameter it comes from, and for every circle it equals π times the diameter, C = πd, or twice π times the radius, C = 2πr. The radius runs from the centre to the edge; the diameter runs straight across through the centre and is twice as long.

π itself is irrational: Johann Lambert proved in 1761 that it cannot be written as a fraction, and Ferdinand von Lindemann showed in 1882 that it is transcendental, which settled that no ruler-and-compass construction can produce a straight line exactly as long as a given circle's boundary. Fractions such as 22/7 and 355/113 are approximations, useful ones, but never the real value; Zu Chongzhi's 355/113 is good to six decimal places, 3.1415929 against 3.1415927.

Circumference and area are bound together. The area A = πr² grows at a rate of exactly 2πr as the radius grows, so the circumference is the derivative of the area with respect to the radius: add a thin ring of width Δr around a circle and you add very nearly C × Δr of area. The same relationship links a sphere's surface area to its volume.

How to use this calculator.

1. 01Choose Radius or Diameter under Input type to match what you measured. A measurement straight across, with calipers or a rule, is a diameter; a distance from a centre point, such as a compass setting or a pivot, is a radius.
2. 02Type the number into Input value in whatever unit you measured — millimetres, inches, metres. The circumference comes back in the same unit, so a diameter in inches gives a circumference in inches.
3. 03On pipes, tubes and rings, decide first which diameter you need: the outside one for anything that wraps around, the inside one for anything that lines the bore. They differ by twice the wall thickness.
4. 04Read the circumference and the diameter beside it, which is a quick check when you started from a radius. Add any overlap, seam or fastening allowance yourself; the result is the bare geometric length.

The formula.

C = 2πr = πd. The two forms are one equation, since d = 2r; use whichever matches the measurement you have. π is a pure number, so the unit of the answer is the unit of the input.

Why should the ratio be the same for every circle? Because all circles are scaled copies of one another: enlarge a circle by any factor and its boundary and its diameter both grow by that factor, so their ratio cannot change. Finding what the ratio actually is takes more work, and Archimedes' polygon method still shows it best. A regular polygon of n sides inscribed in a circle of radius r has perimeter 2nr sin(π/n); one drawn around the outside has perimeter 2nr tan(π/n). The circle's boundary lies between the two. With 96 sides they come to 3.14103 and 3.14271 times the diameter — the basis of his bounds 223/71 and 22/7 — and every doubling of the sides tightens the squeeze.

The constant ratio belongs to flat space. On a curved surface, a circle of radius r measures 2πr(1 − Kr²/6 + …), where K is the surface's curvature. On the Earth, a circle drawn at a ground distance of 1,000 km from its centre point measures 6,257.42 km rather than the 6,283.19 km of flat geometry, about 25.8 km or 0.41% shorter. At a radius of 5 km the shortfall is around 3 mm, a part in ten million, which is why plane geometry serves every building, survey and machine.

Errors in the measurement pass straight through, magnified by π. Misread a diameter by 5 mm and the circumference is out by 15.7 mm; misread a radius by the same 5 mm and it is out by 31.4 mm, because the radius is doubled first.

A worked example.

Frequently asked questions.

References& sources.

1. [1]BIPM (2019). The International System of Units (SI Brochure), 9th ed.
2. [2]NIST SP 811 (2008). Guide for the Use of the International System of Units (SI).
3. [3]Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer.
4. [4]Beckmann, P. (1971). A History of Pi. St. Martin's Press.
5. [5]NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/ (Section 3.12)