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

### Circle Overlap

```
Date: 11/08/2001 at 21:52:11
From: Justin Zaborowski
Subject: Circle Overlap

Both Circle A and Circle B have a radius of 1 unit. The centers of
the two circles are 1 unit apart as well. Find the area of the union
of the two circles.

I really am thankful for all of your help.
```

```
Date: 11/09/2001 at 09:01:42
From: Doctor Mitteldorf
Subject: Re: Circle Overlap

Dear Justin,

Here's a way to think about it.

First, draw two unit equilateral triangles inside the overlapping
area, with their vertices at the centers of the circles and the
intersections of the circles.

Can you see why these triangles must be equilateral?

Next, use Pythagoras to find the area of each equilateral triangle.
You should get an answer of sqrt(3)/4 for each one.

Last, there are four lens-shaped pieces outside the equilateral
triangles. Think of each one of these as the difference between (1/6
of a unit circle) and (a unit equilateral triangle).  In other words,
each lens should have an area of (pi*r^2)/6 - sqrt(3)/4.

Finally, add up the pieces: two equilateral triangles and 4 lens-
shaped differences.

- Doctor Mitteldorf, 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