Diameter

Definition · applies to sets

Elementary

The diameter of the circle is the distance across a circle, measured as the length of a straight line drawn through its center.

High school

Consider a circle or sphere $S$. The diameter of $S$ is the length of every line segment through the center of $S$ with both endpoints on the boundary of $S$.

Undergraduate (metric spaces)

Consider a set $A$ in a metric space with metric $d$. The diameter of $A$ is the largest distance between two points of $A$:

\[\operatorname{diam}(A) = \sup \setdef{d(x, y) \suchthat x, y \in A}.\]

Examples

Worked examples

• The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$ : $2$ High school
  The diameter of the closed unit disk is $2$, the distance between two opposite points of its boundary.
• $[0, 1]$ : $1$ High school
  The diameter of $[0, 1]$ is $1$, the distance between its endpoints.
• $\mathbb{Q} \cap [0, 1]$ : $1$ High school
  The diameter of $\mathbb{Q} \cap [0, 1]$ is $1$, because it contains $0$ and $1$.
• The set of real numbers, $\reals$ : $\infty$ High school
  The diameter of $\reals$ is $\infty$, because $\reals$ contains points arbitrarily far apart.
• The set $\setdef{0, 1}$, as a subset of $\reals$ : $1$ High school
  The diameter of $\setdef{0, 1}$ is $1$, the distance between its two points.
• The Cantor set : $1$ Undergraduate
  The diameter of the Cantor set is $1$, because it contains $0$ and $1$.

Properties

• The diameter of a circle or sphere is twice its radius. In a general metric space, a ball of radius $r$ has diameter at most $2r$, and possibly less.

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