Circle Calculator
Enter any one measurement of a circle, radius, diameter, circumference, or area, and the calculator instantly derives all the others. Switch units freely, add a central angle for sector and arc results, and follow the step-by-step working to see exactly how every answer is reached.
Formula
Worked example
Radius 5 cm: area = pi x 5^2 = 78.5398 cm^2, circumference = 2 x pi x 5 = 31.4159 cm, diameter = 10 cm. For a 90 deg sector: sector area = 19.6350 cm^2, arc length = 7.8540 cm, chord = 7.0711 cm.
How the circle calculator works
Every property of a circle, radius, diameter, circumference, and area, is an exact mathematical function of every other. This calculator accepts any one of those four measurements as input and derives the remaining three in a single step. Switch the "I know the" selector to tell the calculator which measurement you have, enter the value, choose a unit, and all results update instantly. The solve logic inverts the standard formulas: given the area, for example, the radius is the square root of the area divided by pi; given the circumference, it is the circumference divided by 2 pi.
Sector area and arc length
Toggle on "Calculate sector and arc" to unlock three additional outputs for a circular slice. A sector is the pie-slice region bounded by two radii and the arc between them. Its area is one-half times radius squared times the central angle in radians, which is the same as (angle / 360) times the full circle area. The arc length, the curved boundary of the slice, is radius times the angle in radians. The chord length is the straight line connecting the two endpoints of the arc, calculated as 2 times radius times the sine of half the central angle. These are exact Euclidean results with no approximation beyond the value of pi.
Unit switching and exact pi form
Each input field carries a unit selector covering millimetres, centimetres, metres, inches, feet, and yards. All outputs are expressed in the same unit you choose, with area automatically in that unit squared. Linear outputs (circumference, diameter, radius, arc, chord) are also shown as a multiple of pi in parentheses, so you can see the exact symbolic form alongside the decimal approximation. This matches the output style used in textbooks and makes it easier to verify hand calculations or computer-algebra results.
Why area scales with the square of the radius
The area formula A = pi r^2 is a quadratic function of r, meaning the exponent on r is 2. Doubling the radius multiplies the area by 4; tripling it multiplies the area by 9. This quadratic growth is geometrically intuitive: if you scale a shape by a factor k in every direction, its area scales by k^2 because you are multiplying two perpendicular dimensions. Circumference, by contrast, is a linear function of r, so it scales in exact proportion to any change in radius.
Real-world applications
Circle calculations appear across engineering, construction, science, and everyday life. Pipe cross-sections, wheels, circular tanks, pizza sizing, satellite dish coverage, and roundabout road design all rely on the same three formulas. The sector and arc extensions are used in belt-and-pulley design, irrigation pivot coverage, pie-chart data encoding, and any situation where only part of a circle is relevant. For three-dimensional objects such as cylinders, spheres, or cones, these two-dimensional results serve as the building block: the base area of a cylinder is pi r^2, and its lateral surface area is 2 pi r h.
Circle formulas: solve from any known value
| Known | Radius (r) | Diameter (d) | Circumference (C) | Area (A) |
|---|---|---|---|---|
| From r | r | 2r | 2 pi r | pi r^2 |
| From d | d / 2 | d | pi d | pi (d/2)^2 |
| From C | C / (2 pi) | C / pi | C | C^2 / (4 pi) |
| From A | sqrt(A / pi) | 2 sqrt(A / pi) | 2 sqrt(pi A) | A |
r = radius, d = diameter, C = circumference, A = area, theta = central angle in radians.
Frequently asked questions
Can I find the radius if I only know the circumference?
Yes. Set the "I know the" selector to Circumference, enter the value, and the calculator returns the radius directly. Algebraically, radius equals circumference divided by 2 pi, so a circumference of 31.416 cm gives a radius of exactly 5 cm.
How do I calculate the area from the diameter?
Set the selector to Diameter and enter the measurement. The calculator divides by 2 to get the radius, then applies pi times radius squared. You can also do it manually: area equals pi times (diameter / 2) squared, or equivalently pi times diameter squared divided by 4.
What is a sector and how is its area calculated?
A sector is the pie-slice region of a circle bounded by two radii and the arc between them. Its area equals (central angle in degrees / 360) times the full circle area, or equivalently one-half times radius squared times the angle in radians. Toggle on "Calculate sector and arc" and enter the central angle to get sector area, arc length, and chord length.
Does doubling the radius double the circumference?
Yes, circumference scales linearly with radius because the formula C = 2 pi r has r to the power of 1. Doubling r doubles C exactly. Area, however, scales with r squared, so doubling the radius quadruples the area, not doubles it.
Why is pi used in the formulas?
Pi (approximately 3.14159) is the universal ratio of any circle's circumference to its diameter. It is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal expansion never repeats. Because both circumference and area are derived from this ratio, pi appears in both formulas.
What unit should I use for the area output?
The calculator reports area in the squared version of whatever linear unit you choose. If you pick centimetres, area is in square centimetres (cm^2). If you pick inches, area is in square inches (in^2). You do not need to enter a separate unit for area; it follows automatically from the length unit.