Diameter
GeometryThe 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 going from one side of a circle to the other, passing exactly through the center. The diameter is always twice the radius and is the longest possible chord of a circle.
Example
A pizza with a $12$-inch diameter has a $6$-inch radius. If you cut a pizza straight through the center, the cut is a diameter. The diameter of a coin is the widest measurement across it through the center.
Key Insight
The diameter divides a circle into two equal halves called semicircles. Any angle inscribed in a semicircle (with the diameter as its chord) is always a right angle - this is Thales' theorem and it is a beautiful surprise.
Definition
The diameter of a circle is the longest chord, passing through the center with length $d = 2r$. Circumference $= \pi d$. Thales' Theorem: any angle inscribed in a semicircle (where the diameter is the subtended chord) is a right angle. A diameter is both a chord and a line of symmetry of the circle.
Example
Circle with diameter $14$: radius $= 7$, circumference $= 14\pi$ approximately $44.0$, area $= 49\pi$ approximately $153.9$. By Thales' theorem, if $AB$ is a diameter and $C$ is any other point on the circle, then angle $ACB = 90^\circ$.
Key Insight
Thales' theorem (angle in semicircle $= 90^\circ$) is a special case of the inscribed angle theorem: the central angle subtended by a diameter is $180^\circ$, so the inscribed angle is $180/2 = 90^\circ$. It explains why you can find the circumscribed circle of a right triangle: the hypotenuse is always the diameter.
Definition
The diameter is the largest element of the metric space of the circle viewed as a $1$-manifold, and is also the diameter of the closed disk in the sense of metric spaces: $\text{diam}(D) = \sup\{d(x,y) : x,y \in D\} = 2r$. In Riemannian geometry, the diameter of a manifold generalizes this: it is the supremum of geodesic distances.
Example
The diameter of the $n$-sphere $S^n$ in $\mathbb{R}^{n+1}$ is $2$ (for unit sphere). The diameter of a compact Riemannian manifold bounds the lengths of closed geodesics by Bonnet-Myers theorem: if sectional curvature $\ge k > 0$, then $\text{diam} \le \pi/\sqrt{k}$.
Key Insight
The Bonnet-Myers theorem connects curvature to diameter: positive curvature forces compactness and finite diameter. This is how cosmologists used curvature to reason about the finite vs. infinite size of the universe - the geometry of the large-scale universe determines its diameter in the Riemannian sense.