Diameter
The diameter is a chord that passes through the center of a circle, equal to twice the radius and the longest chord in the circle.
Formula
d = 2r; \text{Circumference} = \pi d
Definition
The diameter is a straight line from one side of a circle to the other passing exactly through the center, always twice the radius, $d=2r$, and the longest possible chord of the circle; circumference $=\pi d$. A diameter divides a circle into two semicircles, and by Thales' Theorem, any angle inscribed in a semicircle (with the diameter as its chord) is always a right angle. Formally, the diameter equals the diameter of the closed disk in the metric-space sense, $\text{diam}(D)=\sup\{d(x,y):x,y\in D\}=2r$, a notion that generalizes in Riemannian geometry to the supremum of geodesic distances on a manifold.
Example
A pizza with a $12$-inch diameter has a $6$-inch radius; cutting a pizza straight through the center makes a diameter cut. For diameter $14$: radius $=7$, circumference $=14\pi\approx44.0$, area $=49\pi\approx153.9$, and if $AB$ is a diameter with $C$ any other point on the circle, angle $ACB=90^\circ$ by Thales' theorem. The diameter of the unit $n$-sphere in $\mathbb{R}^{n+1}$ is $2$, and the Bonnet-Myers theorem bounds a manifold's diameter by $\pi/\sqrt{k}$ when its sectional curvature is at least $k>0$.
Key Insight
Thales' theorem, that an inscribed angle on a semicircle is always $90^\circ$, is a special case of the inscribed angle theorem, since the central angle subtended by a diameter is $180^\circ$ and the inscribed angle is always half that; this is also why the hypotenuse of any right triangle is the diameter of its circumscribed circle. The Bonnet-Myers theorem connects curvature to diameter, showing that positive curvature forces compactness and a finite diameter, a principle cosmologists have used to reason about whether the large-scale geometry of the universe is finite or infinite.