Circle Calculator
Enter any one property of a circle — radius, diameter, circumference or area — and get the other three instantly.
How to use this calculator
- 1Pick which property you already have.
- 2Enter its value — the other three are calculated immediately.
- 3The sector table covers the common angles for pie charts and arcs.
How the calculation works
C = 2πr A = πr² d = 2r r = √(A/π)- r
- Radius — centre to edge
- d
- Diameter — all the way across, through the centre
- C
- Circumference — the distance around
- A
- Area enclosed
π is the ratio of circumference to diameter, identical for every circle. It is irrational, so no decimal expansion ever terminates or repeats.
Each formula rearranges to recover the radius, which is why one property determines all four.
Arc length and sector area both use radians, where a full turn is 2π rather than 360.
Worked example
A circle of radius 5 cm
- 1.Diameter = 2 × 5 = 10 cm.
- 2.Circumference = 2π × 5 = 31.4159 cm.
- 3.Area = π × 5² = π × 25 = 78.5398 cm².
Result: Area 78.5398 cm², circumference 31.4159 cm
What a circle is
A circle is the set of every point on a flat plane that sits exactly the same distance from one fixed centre point. That single defining property is what gives a circle its perfect symmetry — unlike almost any other shape, it looks identical no matter which direction you view it from around its centre.
A few terms describe its parts precisely: the radius is the distance from centre to edge, the diameter is the full distance across through the centre (always exactly twice the radius), the circumference is the distance around the outside, and a chord is any straight line connecting two points on the edge without necessarily passing through the centre.
Understanding pi
Every circle, no matter how large or small, has exactly the same ratio between its circumference and its diameter. That ratio is π (pi), and it is irrational — its decimal expansion never terminates and never settles into a repeating pattern, which is why it is written as a symbol rather than a finite number.
Ancient civilisations approximated π using practical measurement long before it could be calculated precisely. One of the earliest rigorous approaches came from Archimedes, who bounded π between the perimeters of many-sided polygons drawn just inside and just outside a circle — squeezing the true value into an ever-narrower range as the polygons gained more sides.
Area versus circumference
Circumference and area answer different questions and scale differently as a circle grows. Circumference is a length, so it grows in direct proportion to the radius — double the radius and the circumference exactly doubles. Area is a two-dimensional quantity, so it grows with the square of the radius — double the radius and the area quadruples, even though the circle only looks "a bit bigger" in a drawing.
This is the same reason a 16-inch pizza carries more than twice the food of a 12-inch one, despite the size difference sounding modest: the relevant comparison is area, not diameter.
Where circle geometry shows up
Circular geometry is everywhere something needs to rotate, roll, or distribute evenly around a centre.
- Wheels and gears — circumference determines how far a wheel travels per rotation, which is the basis for odometers, gear ratios and belt-driven machinery.
- Architecture — domes, arches and rotundas rely on a circle’s structural symmetry to distribute load evenly in every direction.
- Manufacturing — pipes, tanks and turned parts are specified by diameter, and their cross-sectional area determines flow rate or material use.
- Astronomy — orbits are close to circular (more precisely, elliptical) for many practical purposes, and circle geometry gives a first working approximation.
- Sports and tracks — running tracks and field markings use circular arcs and sectors, staggered at the start line to equalise distance across lanes of different radii.
Working with arcs and sectors
A sector is a "slice" of a circle — the pie-shaped region between two radii and the arc connecting them. Because a sector is a fixed fraction of the whole circle, both its arc length and its area scale directly with the fraction of the full 360° turn it covers.
Angles in arc and sector calculations are usually worked in radians rather than degrees, where a full turn is 2π instead of 360 — a unit built directly from the radius, which is what keeps the arc-length and sector-area formulas so simple.
What this assumes, and where it stops
Assumptions
- A perfect circle on a flat plane.
Limitations
- π is irrational, so every result is a decimal approximation — accurate here to about 15 significant figures.
- Ellipses are not circles; their area is πab and their perimeter has no exact closed form. Use the area calculator for those.
Common questions
How do I find the radius from the area?
Divide the area by π, then take the square root: r = √(A/π). For an area of 78.54 cm², that is √(78.54 ÷ 3.14159) = √25 = 5 cm.
Why does doubling the radius quadruple the area?
Because area depends on r², not r. Doubling the radius means squaring twice as much: (2r)² = 4r². This is why a 16-inch pizza has more than twice the food of a 12-inch one despite sounding only a third larger.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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