Circle Calculator

About the circle calculator

A circle is the set of points in a plane equidistant from a central point; that distance is the radius r. The diameter d is twice the radius, the circumference C is the boundary length, and the area A is the flat surface enclosed. These four quantities are bound together by the constant pi, defined as the ratio of any circle's circumference to its diameter (C / d = pi for every circle, regardless of size). Knowing any one of r, d, C, A determines all three others. This calculator handles all four input forms and computes the missing three using the standard Euclidean formulas implemented at IEEE 754 double-precision.

Circles appear in nearly every applied geometry problem: piping and tubing cross-sections, gear and pulley sizing, drum and tank capacity, irrigation circles, pizza and cake sizes, satellite footprints, and the area of mowed lawn under a rotating sprinkler. Knowing how to move quickly between the four quantities saves a step in almost every practical use.

The four formulas

d = 2 r
C = 2 pi r       = pi d
A = pi r^2       = pi d^2 / 4    = C^2 / (4 pi)
r = d / 2        = C / (2 pi)    = sqrt(A / pi)

• Diameter to radius and back. d = 2r so r = d / 2. The diameter passes through the centre; it is always exactly double the radius.
• Circumference. C = 2 pi r approximately equals 6.2832 r. For a circle with radius 1 m, the boundary is roughly 6.28 m around.
• Area. A = pi r squared approximately equals 3.1416 r squared. A circle of radius 1 m has area roughly 3.14 square metres.
• Back-solving from area or circumference. r = sqrt(A / pi) when only the area is given; r = C / (2 pi) when only the circumference is given. The calculator uses these inverse forms automatically when you pick the matching input.

Worked example

A circular garden bed has a measured circumference of 12 metres. Find its radius, diameter, and area.

1. Pick Circumference. Select Circumference in the calculator drop-down and enter 12.
2. Radius. r = C / (2 pi) = 12 / (2 * 3.141593) = 12 / 6.283185 = 1.9099 m.
3. Diameter. d = 2 r = 2 * 1.9099 = 3.8197 m.
4. Area. A = pi r squared = 3.141593 * 1.9099 squared = 3.141593 * 3.6478 = 11.4592 sq m.

Notice the area number (11.46) is almost equal to the circumference number (12.00). That is the famous isoperimetric ratio: among all closed plane shapes with a 12 m perimeter, the circle encloses the largest area, here 11.46 sq m. A square with the same perimeter encloses only 9 sq m.

Quick reference for common radii

Notice that area scales as the square of radius (doubling r quadruples A) while circumference scales linearly. A pizza of 16-inch diameter has twice the area of a 12-inch, not a third more.

Pitfalls to watch for

• Mixing radius and diameter. Many real-world specs (pipe sizes, drill bits, wheels) quote diameter, not radius. Halve before plugging into the radius slot, or pick Diameter in the drop-down.
• Wrong units of area. Area is units squared. A 5 cm radius circle has area 78.54 sq cm, not 78.54 cm. Forgetting the squared unit shows up constantly in homework grading and DIY material orders.
• Approximating pi as 22/7 or 3.14. Those are 0.04 percent and 0.05 percent low respectively. Fine for back-of-envelope; for engineering tolerances under a millimetre on a 1-metre object, use the full Math.PI value.
• Confusing area with surface area of a sphere. The area of a flat circle is pi r squared. The surface area of a sphere is 4 pi r squared (exactly four times larger). Confusing them is the most common geometry mistake.
• Using diameter in the circumference formula by accident. C = pi d, not 2 pi d. Double-counting doubles the perimeter.
• Floating-point rounding on long chains. If you chain area then back-solve to radius and then forward to circumference, you may see the last digit drift by 1 ULP. The calculator shows four decimals to mask this; for high precision compute directly from the original input.

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Frequently asked questions