Perimeter of a Circle Formula – Circumference Guide

The perimeter of a circle formula gives the circumference — the total length of the boundary around a circle.

Quick Reference:

Definition: The circumference is the distance around the outside of a circle — the circular equivalent of perimeter. Symbol: $C$ (circumference) Formula (using radius): $C = 2\pi r$ Formula (using diameter): $C = \pi d$ Constant: $\pi \approx 3.14159$ Type: Measurement formula — geometry Used in: Geometry, engineering, physics, everyday measurement

Full Definition

The perimeter of a circle is called the circumference. Unlike polygons — where perimeter is found by adding straight sides — a circle has a curved boundary, so the formula involves $\pi$, the ratio of a circle's circumference to its diameter.

The two forms of the formula are equivalent: since diameter $d = 2r$, substituting gives $C = \pi d = \pi(2r) = 2\pi r$.

Why π is in The Formula — And Where It Comes From

The constant $\pi$ (pi) arises from a fundamental geometric truth: for any circle, no matter its size, the ratio $\dfrac{C}{d}$ is always the same number — approximately 3.14159. This was known to ancient Babylonian and Egyptian mathematicians and was formalised by Archimedes of Syracuse (c. 287–212 BCE, Sicily), who proved that $\pi$ lies between $\dfrac{223}{71}$ and $\dfrac{22}{7}$ by inscribing and circumscribing polygons around a circle. The symbol $\pi$ was introduced by Welsh mathematician William Jones (1675–1749) in 1706 and popularised by Leonhard Euler (1707–1783, Switzerland) from 1736 onward.

Variable Key

Worked Examples

Example 1: Circumference from radius

A circle has radius $r = 7$ cm. Find the circumference. (Use $\pi \approx \frac{22}{7}$.)

$$C = 2\pi r = 2 \times \frac{22}{7} \times 7 = 2 \times 22 = 44 \text{ cm}$$

Final answer: $C = 44$ cm

Example 2: Circumference from diameter

A circular running track has diameter $d = 100$ m. Find the distance around the track.

$$C = \pi d = 3.14159 \times 100 \approx 314.16 \text{ m}$$

Final answer: $C \approx 314.16$ m

Example 3: Finding radius from circumference

A circle has circumference $C = 88$ cm. Find the radius. (Use $\pi = \frac{22}{7}$.)

$$r = \frac{C}{2\pi} = \frac{88}{2 \times \frac{22}{7}} = \frac{88 \times 7}{44} = \frac{616}{44} = 14 \text{ cm}$$

Final answer: $r = 14$ cm

Common Confusions With The Perimeter of a Circle Formula

Circumference vs area: Circumference ($C = 2\pi r$) is the length of the boundary — a one-dimensional measurement in units of length (cm, m). Area of a circle ($A = \pi r^2$) is the space inside — in square units (cm², m²). The formulas look similar but measure entirely different things.

Radius vs diameter in the formula: $C = 2\pi r$ uses the radius (half the width); $C = \pi d$ uses the diameter (full width). Substituting the diameter into $2\pi r$ gives double the correct answer — the most common calculation error.

$\frac{22}{7}$ vs $\pi$: $\frac{22}{7} \approx 3.1429$ is an approximation of $\pi$, not the exact value. Problems specifying $\pi = \frac{22}{7}$ use this fraction for clean calculation; problems specifying $\pi = 3.14$ or "use $\pi$" expect the symbolic or decimal value.