# Circumference formula of a circle

## Formula

$c \,=\, 2\pi r$

### Introduction

The term circumference is often heard while studying the circles in geometry. It can be calculated mathematically from the radius or diameter of a circle. The mathematical relation that expresses the relation between the circumference of a circle and its radius is called the circumference formula of a circle.

Now, let’s learn the circumference formula of a circle.

1. In geometry, the radius of a circle is usually denoted by a letter $r$.
2. The mathematical constant pi is represented by a Greek symbol $\pi$.
3. Let us assume that the circumference of a circle is denoted by a letter $c$.

The circumference of a circle is equal to two times the product of pi and the radius of the circle.

$c \,=\, 2 \times \pi \times r$

$\implies$ $c \,=\, 2\pi r$

This mathematical formula is called the circumference formula of a circle.

This formula can also be expressed in terms of the diameter of a circle.

$c \,=\, 2 \times \pi \times r$

$\implies$ $c \,=\, \pi \times 2 \times r$

$\implies$ $c \,=\, \pi \times 2r$

In geometry, the diameter of a circle is denoted by a letter $d$.

$\implies$ $c \,=\, \pi \times d$

$\,\,\,\therefore\,\,\,\,\,\,$ $c \,=\, \pi d$

#### Usage

The circumference formula of a circle is mainly used in the following two cases.

1. To calculate the circumference of a circle from its radius.
2. To calculate the radius or diameter of a circle from its circumference.

#### Proofs

The circumference formula of a circle can be derived in two different methods.

##### Geometric Proof

Learn how to derive the circumference formula by geometric approach.

##### Calculus Proof

Learn how to derive the circumference formula by the definite integration method in calculus.

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