Last updated: 2026-05-13

The circle

A circle is defined by a single measurement: any one of radius, diameter, circumference, or area determines all the others. This calculator accepts any of the four as input and computes the remaining three.

The four measurements and their formulas

All four properties are related through the mathematical constant $\pi \approx 3.14159265...$, the ratio of any circle's circumference to its diameter.

Radius ($r$): the distance from the center of the circle to any point on its edge. The radius is the fundamental measurement from which the others are derived.

Diameter ($d$): the distance across the circle through its center — exactly twice the radius:

Circumference ($C$): the distance around the edge of the circle:

Area ($A$): the amount of flat surface enclosed by the circle:

Solving from any starting point

Because all four measurements relate back to the radius, any one of them determines the radius directly:

• From diameter: $r = d/2$, then substitute into the other formulas
• From circumference: $r = C / (2\pi)$
• From area: $r = \sqrt{A / \pi}$

The constant pi

Pi ($\pi$) is the ratio of a circle's circumference to its diameter — a fixed constant for every circle, regardless of size. A circle 1 meter across has a circumference of $\pi$ meters; a circle 100 meters across has a circumference of $100\pi$ meters. Pi is irrational: its decimal expansion (3.14159265358979...) continues infinitely without repeating.

For practical calculations, using $\pi \approx 3.14159$ introduces an error smaller than 0.0001%. This calculator uses full floating-point precision.

Circumference and perimeter

The word "circumference" is simply the term for the perimeter of a circle — the total distance along its boundary. The formula $C = 2\pi r$ gives the exact length of that boundary.

Everyday applications

Engineering and manufacturing — a circular pipe, wheel, or gear is fully specified by any one of its measurements. A wheel with a 35 cm diameter has a circumference of $\pi \times 35 \approx 109.96$ cm — the distance it travels per revolution.

Construction and landscaping — computing the area of a circular garden bed, pool, or foundation requires $A = \pi r^2$. A circular patio with a 4-meter radius covers about 50.3 square meters.

Packaging — the area of a circular label or lid, and the circumference needed for a band or seal, both follow directly from the radius or diameter.

Navigation and mapping — circles of given radius appear in search-and-rescue planning, range calculations, and signal coverage estimations. The area within a 10 km radius is $\pi \times 10^2 \approx 314$ square kilometers.

Cooking — pan sizes are specified by diameter; the area allows recipes to be scaled between pan sizes proportionally ($A \propto r^2$, so doubling the radius quadruples the area).

Sectors and arcs (beyond this calculator)

A sector is a "pie slice" of a circle — a portion bounded by two radii and an arc. Its area is $A_{\text{sector}} = \tfrac{\theta}{2\pi} \cdot \pi r^2 = \tfrac{1}{2}r^2\theta$ where $\theta$ is the central angle in radians. An arc's length is $L = r\theta$. These are derived from the same circle formulas and require the angle as an additional input.