Area of a Reuleaux Triangle

Date: 06/13/2002 at 22:32:02
From: Terry Baume
Subject: area of a Reuleaux triangle


Could you please help me find a formula to find the area of a 
Reuleaux triangle?

Date: 06/15/2002 at 00:20:01
From: Doctor Douglas
Subject: Re: area of a Reuleaux triangle


Hi, Terry.

Thanks for submitting your question to the Math Forum.

Suppose the diameter of the Reuleaux triangle is D.
Then you can think of the entire shape as being constructed
from an equilateral triangle of side length D, and three
segments of a circle (see the following page in our FAQ:
  http://mathforum.org/dr.math/faq/formulas/faq.circle.html ).

Now, you can use the formula for the area of an equilateral
triangle:
 
<http://mathforum.org/dr.math/faq/formulas/
faq.triangle.html#equilateral>,

and add to it the formula for the area of three segments (formula
found on the circle FAQ page above).  You'll need to use the fact that
each angle in an equilateral triangle measures 60 degrees.

I hope this gives you enough information to derive your formula,
but if you're still stuck, please write back!

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/