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Area of a Crescent

Date: 11/10/2000 at 16:04:19
From: Dennis Graham
Subject: Formula for area of a crescent

I am an 8th grade math teacher. The elementary art teacher asked me 
for a formula for the area of a crescent. They were working on 
geometric shapes and discussing area. I have not found anything as of 

Date: 11/10/2000 at 22:46:18
From: Doctor Peterson
Subject: Re: Formula for area of a crescent

Hi, Dennis.

The answer, like any formula, will depend on what dimensions they know 
about the crescent. I'll suppose that they are just asking for a 
general formula, using whatever numbers make it easiest.

But I can't think of any way to define it so as to give a really 
simple formula. Here's one approach:

                 *********** C
             ****           **oooooooooo
          ***            ooo/ \ ***.....oooo
        **             oo  /   \   **.......oo
       *             oo   /     \    *........oo
      *             o    /r1    \r2   *.........o
      *             o   /        \    *.........o
     *             o   /          \    *.........o
     *             o  A\     d    /B   *....d....o
      *             o   \        /    *.........o
      *             o    \      /     *.........o
       *             oo   \     /    *........oo
        **             oo  \   /   **.......oo
          ***            ooo\ / ***.....oooo
             ****           **oooooooooo
                 *********** D

The area of the shaded crescent, bounded by arcs of a circle with 
center A and radius r1, and another circle with center B and radius 
r2, can be found by adding quadrilateral ADBC and sector BDFC, then 
subtracting sector ADEC.

You can find the sector areas if you know the angles DBC and DAC; to 
do that I would use triangle ABC and double the angles. The area of 
this triangle also gives you half the area of ADBC, and can be found 
using Heron's formula with r1, r2, and d. So it can be done, but I'm 
not interested in writing it out as a single formula.

- Doctor Peterson, The Math Forum
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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