Circumference Calculator

The circumference of a circle is C = 2 × π × r, where r is the radius and π is the ratio of any circle's circumference to its diameter — a fundamental constant of geometry. Equivalently, C = π × d using the diameter. The fact that this ratio is the same for every circle (regardless of size) is exactly what defines π and is the reason a single formula handles circles from atomic to astronomical scales.

Going from area to circumference takes one extra step. Area is A = π × r², so r = √(A / π). Plug that into C = 2 × π × r and you get the shortcut C = 2 × √(π × A). A circle with area 78.54 has radius about 5 and circumference about 31.42.

A standard 700c road bike wheel has a tire diameter of about 670 mm (about 26.4 in). Circumference = π × 670 ≈ 2,105 mm, or 2.105 meters per revolution. At a cadence of 90 revolutions per minute in a gear ratio of 1:1, the wheel covers 90 × 2.105 = 189.5 meters per minute — about 11.4 km/h. Cyclometers use this exact calculation, which is why entering a wrong tire size produces noticeable speed and distance errors over a ride.

The inner lane of a standard outdoor running track is a 400-meter loop with two 84.39-meter straights and two semicircular ends with a radius of 36.50 meters. Each curve contributes π × 36.50 ≈ 114.67 meters, so the total perimeter is 2 × 84.39 + 2 × 114.67 = 398.12 meters along the kerb plus a small calibration offset, giving the official 400-meter measurement. Outer lanes are longer (each lane adds about 7.66 meters), which is why staggered starts exist.

Mixing radius and diameter. The single most common error is plugging diameter into the formula C = 2 × π × r. If you only know the diameter, either divide by 2 first or use C = π × d.

Forgetting the unit. Circumference is a length, so the unit of the answer is the unit of the radius — not squared, not cubed. If r is in centimeters, C is in centimeters.

Confusing circumference with perimeter. Both measure the boundary of a 2-D shape. 'Perimeter' is the general term and applies to any polygon; 'circumference' is reserved for circles and ellipses. For an ellipse, circumference has no simple closed form and is usually approximated.

Treating π as 3.14. Two decimals are fine for back-of-the-envelope work but introduce a 0.05% error. For engineering, surveying, or anything requiring more than three significant figures, use π to at least six decimals. This calculator uses JavaScript's Math.PI, which is accurate to about 15 decimal places.