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/