Definition

The circumference is the distance all the way around a circle, the circle's version of perimeter: $C = \pi d$, where $d$ is the diameter, or equivalently $C = 2\pi r$ for radius $r$. The constant ratio $C/d = \pi$ holds for every circle, geometrically defining $\pi$ itself, and this same formula 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. As an 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 formula changes: on a sphere of curvature $K$, $C = 2\pi r_g$ where $r_g$ is the geodesic radius, with correction terms of order $O(r^3)$ for small $r$.