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
Symbol | Meaning | Unit |
|---|---|---|
$C$ | Circumference (perimeter of the circle) | length (cm, m, etc.) |
$r$ | Radius — distance from centre to edge | length |
$d$ | Diameter — distance across the circle through the centre; $d = 2r$ | length |
$\pi$ | Pi — the ratio $C/d$, approximately 3.14159 | dimensionless constant |
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.
Was this article helpful?
Your feedback helps us write better content