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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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