Circle Calculator

Every measurement of a circle comes from its radius, the distance from the centre to the edge. The diameter is twice the radius, the circumference is the distance round the edge, and the area is the space inside. π (pi) is the same for every circle: the circumference divided by the diameter, about 3.14159.

The formula

diameter d = 2 × r
circumference C = 2 × π × r = π × d
area A = π × r²
radius from C: r = C ÷ (2 × π)
radius from A: r = √(A ÷ π)
radius from chord c and height h: r = c² ÷ (8 × h) + h ÷ 2
arc length s = r × θ (θ in radians) = θ° ÷ 360 × 2 × π × r
sector area = θ ÷ 2 × r² = θ° ÷ 360 × π × r²
chord c = 2 × r × sin(θ ÷ 2)
segment height h = r × (1 − cos(θ ÷ 2))
segment area = r² ÷ 2 × (θ − sin θ)
ring area = π × (R² − r²)

Worked example

A circle with a radius of 5 cm: the diameter is 2 × 5 = 10 cm. The circumference is 2 × π × 5 = 10π cm, which is 31.4159 cm to 6 significant figures. The area is π × 5² = π × 25 = 25π cm², which is 78.5398 cm².

In Advanced, the default slice is 60°. 60° is 60 × π ÷ 180 = π/3 radians, about 1.0472. The arc is 5 × π/3 = 5π/3 cm ≈ 5.23599 cm, the sector area is π/3 ÷ 2 × 25 = 25π/6 cm² ≈ 13.09 cm², the chord is 2 × 5 × sin 30° = 2 × 5 × 0.5 = 5 cm, and the segment area is 25 ÷ 2 × (1.0472 − sin 60°) = 12.5 × (1.0471976 − 0.8660254) = 12.5 × 0.1811722 = 2.26465 cm². The default ring with an inner radius of 3 cm has an area of π × (25 − 9) = 16π cm² ≈ 50.2655 cm².

Working backwards

From a circumference of 31.4159 cm the radius is 31.4159 ÷ (2 × 3.14159) = 5 cm. From an area of 78.5398 cm² it is √(78.5398 ÷ 3.14159) = √25 = 5 cm. From a chord of 8 cm with an arc 2 cm high, r = 8² ÷ (8 × 2) + 2 ÷ 2 = 4 + 1 = 5 cm. That last one is handy when you can only measure part of a circle, such as an arch, a curved flower bed or a piece of a broken plate.

Answers in terms of π

Leaving π in the answer keeps it exact. A radius of 5 gives C = 10π and A = 25π. If you start from a plain circumference, the radius has π underneath, e.g. C = 31.4 gives r = 15.7/π. Starting from a plain area gives a square root, e.g. A = 50 gives r = √(50/π). You can also type π into the box: a circumference of 10π gives a radius of exactly 5.

Degrees and radians

A full turn is 360° or 2π radians, so 1 radian is 180 ÷ π ≈ 57.2958°. Radians make the arc formula neat: arc = radius × angle. Type angles in radians as decimals (1.0472) or with π (π/3).

Sector, segment and chord

• A sector is a slice from the centre, like a slice of pizza. Its perimeter is the arc plus two radii.
• A chord is a straight line joining two points on the edge. The longest chord is the diameter.
• A segment is the piece cut off by a chord: the sector minus the triangle. Its height (the sagitta) is measured from the middle of the chord to the arc.
• Angles over 180° give the bigger (major) piece. A chord gives the smaller piece; the calculator shows the rest of the circle as well.

Units

Areas change with the square of the length unit: 1 m = 100 cm, so 1 m² = 10,000 cm². 1 ft = 12 in, so 1 ft² = 144 in². 1 acre is 43,560 ft² and 1 hectare is 10,000 m². You can type in one unit and see answers in another.

π used here is JavaScript’s built-in value, 3.141592653589793. For triangles, use the scientific calculator; for floor and plot areas in square feet or metres, try the square footage calculator.