Circle Calculator: Radius, Diameter, Area, Circumference

Circle calculator

A circle calculator finds the radius, diameter, circumference, and area of a circle from any one of those four values. Enter a single known measurement and the calculator computes the other three using standard geometry formulas.

A circle is the set of all points in a plane that lie at a fixed distance from a center point. That fixed distance is the radius. The diameter, the circumference, and the area all follow from the radius alone.

How to use the calculator

1. Choose the known parameter: radius, diameter, circumference, or area.
2. Enter its numeric value. The value must be zero or greater; negative lengths and areas are not physically meaningful.
3. The calculator displays the radius, diameter, circumference, and area, each rounded to two decimal places.

The calculator also draws the circle. The drawing is a schematic, not to scale, so a tiny circle and a huge one look about the same size.

Entering zero is allowed and returns zero for all four measurements, since a circle of radius zero is a single point. Non-numeric entries produce a validation error instead of a result.

Circle formulas

Let r stand for radius and d for diameter. The constant π (pi) is approximately 3.14159; it is the ratio of a circle's circumference to its diameter.

Diameter from radius:

$d = 2r$

Circumference:

$C = 2\pi r = \pi d$

Area:

$A = \pi r^2 = \frac{\pi d^2}{4}$

Radius from circumference:

$r = \frac{C}{2\pi}$

Radius from area:

$r = \sqrt{\frac{A}{\pi}}$

From any single value, the remaining three follow by first solving for the radius, then applying the diameter, circumference, and area formulas above.

How to calculate circle measurements from each starting value

Starting from radius (r):

• Diameter: $d = 2r$
• Circumference: $C = 2\pi r$
• Area: $A = \pi r^2$

Starting from diameter (d):

• Radius: $r = \frac{d}{2}$
• Circumference: $C = \pi d$
• Area: $A = \frac{\pi d^2}{4}$

Starting from circumference (C):

• Radius: $r = \frac{C}{2\pi}$
• Diameter: $d = \frac{C}{\pi}$
• Area: $A = \pi r^2$

Starting from area (A):

• Radius: $r = \sqrt{\frac{A}{\pi}}$
• Diameter: $d = 2r$
• Circumference: $C = 2\pi r$

Worked example

A circle has a radius of 5 units.

• Diameter: d = 2 × 5 = 10 units.
• Circumference: C = 2π × 5 ≈ 31.42 units.
• Area: A = π × 5² ≈ 78.54 square units.

The same circle can be described by its circumference of about 31.42 units. Dividing 31.42 by 2π returns 5.00 units after rounding, so the radius comes back, and the diameter and area follow as before. Any one of the four values fixes the other three.

Circle formulas in geometry

The number π is irrational, meaning its decimal digits never terminate or repeat. Ancient mathematicians approximated it: the Babylonians and Egyptians used rough fractions, and Archimedes of Syracuse, around 250 BCE, showed that π lies between 223/71 and 22/7 by using polygons drawn inside and outside a circle. The symbol π was popularized in the 18th century by the mathematician Leonhard Euler. Modern calculators use π to many more decimal places than any physical measurement requires.

Circle formulas are used to size pipes, gears, and wheels, to work out how much material a round structure needs, and to model orbits in astronomy.

Frequently asked questions

How is the circumference of a circle calculated? Multiply the radius by 2π, or multiply the diameter by π: C = 2πr = πd. A circle with a 5-unit radius has a circumference of about 31.42 units.

What is the formula for the area of a circle? Area equals π times the radius squared: A = πr². A circle with a 5-unit radius has an area of about 78.54 square units.

How do I find the radius from the circumference? Divide the circumference by 2π (about 6.2832). A circumference of 31.42 units gives a radius of about 5 units.

How do I find the radius from the area? Divide the area by π, then take the square root: r = √(A / π). An area of 78.54 square units gives a radius of about 5 units.

What is the relationship between diameter and radius? The diameter is always twice the radius (d = 2r), and the radius is always half the diameter (r = d / 2).

Can circle measurements use any unit? Yes. Radius, diameter, and circumference share the same length unit (such as centimeters or inches), and area is expressed in that unit squared. The calculator does not convert between units; results use whatever unit was entered.

References

1. "Circle." Wolfram MathWorld. https://mathworld.wolfram.com/Circle.html
2. "Circumference and Area of a Circle." Khan Academy. https://www.khanacademy.org/math/basic-geo/basic-geo-circles
3. Beckmann, Petr. A History of π. St. Martin's Press, 1971.
4. Archimedes. Measurement of a Circle. https://www.math.ubc.ca/~vjungic/students/Archimedes-Measurement%20of%20a%20Circle.pdf