Circle Calculator

How to use the circle calculator

Choose which measurement you have, type it, and pick the unit it is in. The calculator returns all four measurements of the circle:

• the radius, from the center to the edge;
• the diameter, straight across through the center, twice the radius;
• the circumference, the distance around;
• the area inside, in square units.

When you start from the radius or diameter, the circumference and area are also given as exact multiples of π, the form most math classes ask for: a radius of 5 gives a circumference of 10π and an area of 25π. From a circumference or an area the radius itself involves π, so only decimals are shown.

Add a central angle to measure a slice, or sector: its arc length along the edge, its area, the straight chord across its ends and the segment between chord and arc. The angle can be in degrees or radians; switching converts the number you typed.

Circle formulas

r

radius

d

diameter

C

circumference

A

area

π

the ratio of any circle’s circumference to its diameter, about 3.14159265

Solved for the radius, so any one measurement gives the rest:

For a sector with central angle θ in radians (degrees × π ÷ 180):

The sector is the fraction θ ÷ 2π of the whole circle, which is why its arc and area are that fraction of C and A. The segment is the sector minus the triangle formed by the two radii and the chord, whose area is r² sin θ ÷ 2. Past 180° that triangle lies outside the sector, so the segment is larger than the sector.

The calculator uses the double-precision value of π built into JavaScript, the closest such number to π, which is accurate to about 16 significant digits, far more than any measurement needs.

Worked examples

Radius 5 cm (the default). The diameter is 2 × 5 = 10 cm. The circumference is 2π × 5 = 10π ≈ 31.4159265359 cm, and the area π × 5² = 25π ≈ 78.5398163397 cm².

A 60° slice of it. 60° is π ÷ 3 ≈ 1.0471975512 radians, one sixth of the circle. The arc is 5 × 1.0471975512 ≈ 5.235987756 cm and the sector area 5² × 1.0471975512 ÷ 2 ≈ 13.08996939 cm². The chord is 2 × 5 × sin 30° = 5 cm, equal to the radius, because the two radii and the chord form an equilateral triangle. The segment between chord and arc is about 2.2646518427 cm².

From a measured circumference. A tape around a tree trunk reads 100 cm. The radius is 100 ÷ 2π ≈ 15.9154943092 cm, so the trunk is about 31.83 cm across and its cross-section about 795.77 cm².

One 12-inch pizza or two 8-inch ones? Pizza sizes are diameters. One 12-inch pizza is π × 12² ÷ 4 = 36π ≈ 113.1 in²; two 8-inch pizzas are 2 × 16π = 32π ≈ 100.5 in². The single large one has more.

Tips and common mistakes

• Radius or diameter? Pipe, wheel, pan and pizza sizes are usually diameters. Using a diameter as the radius makes the area four times too large.
• Square the radius, not the product. πr² means π × r × r; (πr)² is a different, larger number.
• Degrees or radians. The arc and sector formulas need θ in radians. Entering 60 with radians selected is almost ten full turns, which the calculator rejects.
• Area units. Doubling the radius doubles the circumference but multiplies the area by four. To express the area in another unit, use the area converter; for the diagonal or side of a right triangle, the Pythagorean theorem calculator.

Limits

Values can be from just above 0 up to 10¹⁵ in the chosen unit, and the central angle up to one full turn. The π forms are shown only when you enter the radius or diameter and the coefficient has at most 16 digits. The formulas assume a perfect circle on a flat surface; measured objects are rarely perfectly round, so round the results to the precision of your measurement.