Associated Topics || Dr. Math Home || Search Dr. Math

### 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
yet.
```

```
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--+------------+---*E--------oF
*             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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/