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### Perimeter of a Reuleaux Triangle

```
Date: 04/15/2001 at 09:49:51
From: deborah crisp
Subject: Perimeter of a Reuleaux triangle

How can I find perimeter of a Reuleaux triangle of width d? Is the
formula circumference = pi times width (d)? This seems too simple.
```

```
Date: 04/17/2001 at 14:26:30
From: Doctor Douglas
Subject: Re: Perimeter of a Reuleaux triangle

Hi Deborah, and thanks for writing.

Yes, it does seem simple, but there is a theorem that states that all
curves of constant width d have the same perimeter, pi*d: Barbier's
theorem.

You can verify this theorem directly for the Reuleaux triangle.
Suppose that the diameter of the Reuleaux triangle is d. Each side of
the base equilateral triangle has length d. Then the perimeter of the
Reuleaux triangle is composed of three circular arcs, each of which
has arc length equal to

s = radius * subtended_angle
=   d    *    pi/3             angle is measured in radians

and the total perimeter is

S = 3*s = d*pi

I hope this helps answer your question.

- Doctor Douglas, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Euclidean Geometry
High School Euclidean/Plane Geometry

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