Circumference
Geometry & MeasurementCircumference is the distance around the outside of a circle, calculated as pi times the diameter or two times pi times the radius.
Formula
C = 2\pi r = \pi d
Definition
The circumference is the distance all the way around a circle. It is the circle's version of perimeter. The formula is $C = \pi d$, where $d$ is the diameter (the distance across the circle through the center).
Example
A circle with diameter $10$ cm has circumference $C = 3.14 \times 10 = 31.4$ cm. A tire with radius $30$ cm travels $C = 2 \times 3.14 \times 30 = 188.4$ cm in one full turn.
Key Insight
No matter how big or small a circle is, its circumference divided by its diameter always equals pi (about $3.14159$). Pi is what makes circles special among all shapes.
Definition
The circumference of a circle with radius $r$ is $C = 2\pi r$, or equivalently $C = \pi d$ where $d = 2r$ is the diameter. This is the perimeter of the circle, and the constant ratio $C/d = \pi$ holds for every circle, defining pi geometrically.
Example
A circular track has diameter $400$ m: $C = \pi \times 400 = 1256.6$ m per lap. If a cyclist rides $10$ laps, the distance is $12{,}566$ m $= 12.57$ km.
Key Insight
The circumference formula also gives the arc length of a full circle. For a partial arc of central angle $\theta$ (in radians), the arc length is $r\theta$, a direct proportional slice of the circumference.
Definition
The circumference of a circle of radius $r$ in the Euclidean plane is $C = 2\pi r$, derived as the arc length integral: $$C = \int_0^{2\pi} \sqrt{(r\sin t)^2 + (r\cos t)^2} \, dt = r \cdot 2\pi.$$ In non-Euclidean geometry, the circumference formula changes: on a sphere of curvature $K$, $C = 2\pi r_g$ where $r_g$ is the geodesic radius, and for small $r$ the correction is $O(r^3)$.
Example
On a sphere of radius $R$, the circumference of a small circle at geodesic distance $r$ from the pole is $C = 2\pi R \sin(r/R)$, which approaches $2\pi r$ as $r/R$ approaches $0$, recovering the flat result.
Key Insight
The deviation of circumference from $2\pi r$ in curved space is related to the Gaussian curvature by the Bertrand-Diguet-Puiseux theorem. Measuring this deviation is how physicists detect the curvature of spacetime near massive objects.