What is the formula for the area of a circle?

The area of a circle is A = pi times r squared, where r is the radius and pi is approximately 3.14159. For example, a circle with radius 5 cm has an area of pi x 25 = 78.54 cm squared. If you know the diameter instead of the radius, use A = pi x (d/2) squared, or equivalently A = pi x d squared divided by 4.

What is the formula for the circumference of a circle?

The circumference of a circle is C = 2 x pi x r, where r is the radius. This is equivalent to C = pi x d, where d is the diameter. For example, a circle with radius 5 cm has a circumference of 2 x pi x 5 = 31.42 cm. The circumference is also called the perimeter of the circle.

How do I find the radius if I know the circumference or area?

If you know the circumference C, the radius is r = C divided by (2 x pi). If you know the area A, the radius is r = the square root of (A divided by pi). For example, if the circumference is 31.416 cm, the radius is 31.416 divided by 6.2832 = 5 cm. If the area is 78.54 cm squared, the radius is the square root of (78.54 divided by 3.14159) = the square root of 25 = 5 cm.

Circle Formula Calculator

This calculator works out every key measurement of a circle - radius, diameter, circumference and area - the moment you enter just one of them. Geometry problems, trade quotes, DIY projects and maths homework often give you a single circle measurement and ask you to find the rest, and rather than juggling four separate formulas, this tool does the algebra for you. Choose which value you already know from the dropdown - radius, diameter, circumference or area - then type in that figure and pick a length unit such as centimetres, metres, inches or feet for display purposes only, since no unit conversion is applied. The calculator instantly returns all four measurements, radius, diameter, circumference and area, each shown alongside the exact formula used to derive it, such as C = 2πr or A = πr², so you can follow the working rather than just trust the number. You can also choose how many decimal places to show, from whole numbers up to six decimal places, and a worked example using the default radius of 5 cm walks through every step. A summary table below the results lists all eight formulas for converting between radius, diameter, circumference and area, making this a handy reference as well as a calculator, built on standard Euclidean geometry using pi to full floating-point precision.

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Standard geometry All formulas use the mathematical constant pi (π = 3.14159265358979).

1. Known Value

What do you know?

Radius value

e.g. 5 cm, 10 m, 3 inches

2. Unit Label (optional, for display only)

Length unit

Decimal places in results

Circle Measurements

Radius (r)
5.00
given
cm

Diameter (d)

10.00

d = 2r

Circumference (C)

31.42

C = 2πr

Area (A)

78.54

A = πr²

cm²

Worked Example (default inputs)

Input: Radius r = 5 cm

Diameter: d = 2 × r = 2 × 5 = 10.00 cm

Circumference: C = 2 × π × r = 2 × 3.14159 × 5 = 31.42 cm

Area: A = π × r² = 3.14159 × 25 = 78.54 cm²

Circle Formulas Summary

To find	Known	Formula
Radius	Diameter	`r = d / 2`
Radius	Circumference	`r = C / (2π)`
Radius	Area	`r = √(A / π)`
Diameter	Radius	`d = 2r`
Circumference	Radius	`C = 2πr`
Circumference	Diameter	`C = πd`
Area	Radius	`A = πr²`
Area	Diameter	`A = πd² / 4`

Circle Formulas Explained

A circle is a perfectly round shape where every point on the boundary (the circumference) is the same distance from the centre. That distance is called the radius. The diameter is simply twice the radius: the distance from one side of the circle to the other passing through the centre.

All four main measurements of a circle are interconnected. Once you know any one of them, you can calculate the other three using straightforward formulas built around the mathematical constant pi (π = 3.14159265...).

The Four Key Circle Measurements

Measurement	Symbol	Description
Radius	r	Distance from the centre to the edge
Diameter	d	Distance across the circle through the centre (d = 2r)
Circumference	C	The total length around the circle (the perimeter)
Area	A	The total space enclosed inside the circle

Area of a Circle

The area formula A = πr² comes from integrating concentric rings from the centre outward, or from the limit of inscribed polygons with increasing numbers of sides. For practical purposes, multiply pi (approximately 3.14159) by the square of the radius. If you have the diameter, use A = πd² / 4, since r = d/2.

Circumference of a Circle

The circumference C = 2πr is equivalent to C = πd. This relationship defines pi itself: pi is the ratio of any circle's circumference to its diameter, and it is the same constant for every circle regardless of size. To find the circumference of a very large circle such as the Earth (radius approximately 6,371 km), you would calculate 2 × π × 6,371 = 40,030 km.

Solving from Circumference or Area

To find the radius from the circumference, rearrange C = 2πr to get r = C / (2π). To find the radius from the area, rearrange A = πr² to get r = √(A / π). In both cases, once you have the radius all other measurements follow from the standard formulas above.

Units

The radius, diameter, and circumference all share the same length unit (centimetres, metres, inches, etc.). Area is always expressed in square units: if the radius is in centimetres, the area is in square centimetres (cm²). This calculator does not perform unit conversions; it uses whatever unit you enter and labels the area result with the corresponding squared unit.

Related Calculators

Method and sources: Standard Euclidean geometry. Formulas are derived from the definition of pi as the ratio C/d and from integration of the circle's area. Reference: NCEA Level 1 Mathematics (measurement and geometry strand), NZQA; Euclid's Elements, Book III.

This calculator uses the mathematical constant pi to full double-precision floating-point accuracy (approximately 15 significant figures). Results are displayed to your chosen number of decimal places.