Circle Calculator: Area, Circumference, Radius

How to use the calculator

Choose which measurement you know: radius, diameter, circumference or area.
Enter its value. Any unit works: inches give the area in square inches, feet give square feet.
Optionally enter a central angle in degrees (90° by default) for the arc, sector and chord. Clear the field if you don't need them.

The exact π forms (such as 25π) appear only when you start from the radius or diameter. From a circumference or area, finding the radius means dividing by π, so the results are shown as decimals only.

Formulas used

Everything goes through the radius r first:

From the diameter: r = d ÷ 2
From the circumference: r = C ÷ (2π)
From the area: r = √(A ÷ π)

Then d = 2r, C = 2πr and A = πr². For a central angle θ, converted to radians (θ × π ÷ 180):

Arc length = r·θ
Sector area = r²·θ ÷ 2
Chord length = 2r·sin(θ ÷ 2), the straight line between the ends of the arc

The calculator uses π in double precision, 3.141592653589793, the value of JavaScript's Math.PI, and displays it as 3.141592654. It never rounds π to 3.14, so results differ slightly from hand calculations that do. Fractions such as 22/7 are only approximations of π.

Worked examples

1. Which pizza is the better deal?

Pizzas are usually sized by diameter, but what you eat is area. Enter each size as a diameter in inches:

Pizza	Area
8-inch	50.27 in²
10-inch	78.54 in²
12-inch	113.10 in²
14-inch	153.94 in²
16-inch	201.06 in²
18-inch	254.47 in²

Two 8-inch pizzas add up to about 100.53 in², less than one 12-inch. Two 12-inch pizzas, about 226.19 in², beat one 16-inch but not one 18-inch. Divide each price by its area to compare cost per square inch.

2. One slice

A 14-inch pizza cut into 8 equal slices gives each slice a 45° angle. With diameter 14 and angle 45, each slice has an area of 19.242255 in² and 5.497787 in of crust along the edge. The straight cut across the slice's outer end would be 5.357568 in.

3. Diameter from a tape around a tree

A tape wrapped around a trunk reads 47 in. Enter it as the circumference: r = 47 ÷ (2 × 3.141592654) = 7.480282 in, so the diameter is 14.960565 in, about 15 inches.

4. Sizing a round patio

To cover 100 ft², enter the area: the radius is 5.641896 ft and the diameter 11.283792 ft, with 35.449077 ft of edging around it.

Frequently asked questions

What are the formulas for the area and circumference of a circle?

Area = πr² and circumference = 2πr, or πd. For a radius of 5: area 25π ≈ 78.539816 and circumference 10π ≈ 31.415927.

How do I find the radius from the circumference?

Divide the circumference by 2π. A circumference of 47 gives a radius of 7.480282 in the same unit.

How do I find the diameter from the area?

Divide the area by π, take the square root to get the radius, then double it. An area of 100 gives a diameter of 11.283792.

Why does a 60° angle give a chord equal to the radius?

The two radii and the chord form a triangle with a 60° angle between two equal sides, which makes it equilateral. With a radius of 10 and 60°, the chord is exactly 10 and the arc 10.471976, slightly longer.

What happens with a 360° angle?

The sector becomes the whole circle: for a radius of 10, the sector area equals the full area, 314.159265, the arc equals the circumference and the chord is 0.

Does doubling the diameter double the area?

No, it multiplies it by four, because the area depends on the radius squared. A 16-inch pizza has 201.06 in², four times the 50.27 in² of an 8-inch.

Sources

Results are computed in floating-point arithmetic and rounded for display. Real objects are rarely perfect circles, so measured results are estimates.