---
tag: B4YD1ZXRVH
kind: concept
name: Curve of constant width
msc: [52A10]
specializes: [045YFKRW0P]
sources:
  - url: https://en.wikipedia.org/wiki/Curve_of_constant_width
same_as:
  wikidata: Q1192156
  wikipedia: Curve of constant width
---

## Summary

A curve of constant width is the boundary of a plane [[convex body|437FSG3419]] with this property.
Every two parallel supporting lines of the body are at the same distance, called the width.
The circle is the simplest example, but there are many others, such as the Reuleaux triangle.
Curves of constant width are a classical source of extremal problems in plane convex geometry.
All of them with a given width have the same length, but their areas differ.

## Explanations

### L1 human

Put the shape between two parallel rulers that touch it on opposite sides.
If the distance between the rulers is the same however the shape is turned, the boundary of the shape is a curve of constant width.
A coin with this property could roll smoothly between two boards, which is why such shapes can serve as rollers.
A circle of radius $1$ has constant width $2$.
A square does not: its width is $1$ across the sides and $\sqrt{2}$ across the corners.