Circle Calculator

A circle is the set of all points in a plane that are the same distance from a fixed point, its center. Because every measurement of a circle depends only on that one distance, knowing any single measurement is enough to find all the others.

Parts of a circle

The radius is the distance from the center to any point on the circle. The diameter is a straight line through the center from one side to the other, so it is twice the radius. The circumference is the distance around the circle, its perimeter. The area is the flat region it encloses.

Other named parts describe lines and regions related to the circle. A chord is any straight segment joining two points on the circle; the diameter is the longest possible chord. A secant is a line that crosses the circle at two points, extending beyond it, while a tangent touches the circle at exactly one point and is perpendicular to the radius there. An arc is a piece of the circumference between two points, and its length is the radius times the angle it subtends in radians.

A sector is the wedge between two radii and the arc that joins them, like a slice of pie, and its area is ½R²θ for an angle θ in radians. A segment is the region between a chord and its arc. Two circles with the same center but different radii are called concentric, and the ring between them is an annulus, whose area is π(R² − r²).

The constant π

π (pi) is the ratio of any circle's circumference to its diameter. It is the same for every circle, about 3.14159265358979. π is irrational, meaning it cannot be written exactly as a fraction and its decimal digits never end or settle into a repeating pattern. It is also transcendental, which proved in 1882 that the ancient problem of squaring the circle with compass and straightedge is impossible.

Approximations have been used for thousands of years. Babylonian scribes used 3 1/8, and Archimedes trapped π between 3 10/71 and 3 1/7 by drawing polygons with 96 sides inside and outside a circle. The fraction 22/7 is a common classroom approximation, accurate to about 0.04%, and 355/113, found by the Chinese mathematician Zu Chongzhi in the fifth century, is accurate to seven digits.

Computers have now calculated π to well over a hundred trillion digits, though about 40 digits are enough to compute the circumference of the observable universe to within the width of a hydrogen atom. This calculator uses the full double-precision value. π also appears far from circles, in probability, waves, the normal distribution and Euler's identity, e^(iπ) + 1 = 0.

Circle formulas

All four measurements follow from the radius. To go the other way, solve for R: R = D ÷ 2, R = C ÷ (2π), or R = √(A ÷ π). For an area of 100, the radius is √(100 ÷ π) ≈ 5.6419.

D = 2R C = 2πR = πD A = πR² = πD² ÷ 4

R

radius

D

diameter

C

circumference

A

area

π

≈ 3.14159

Frequently asked questions