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What is the Area Not Shared by the Circles?


Date: 3 Mar 1995 10:01:35 -0500
From: Mary Basse
Subject: Geometry

When I was taking the SAT test, I ran across this problem. Will 
you please help me? Two circles intersect such that their centers 
and their points of intersection form a square with each side 
equal to 3.  What is the total area of the sections of the square 
that are not shared by both circles?

Thank you.
Russell

mbasse@isbe.state.il.us


Date: 3 Mar 1995 10:41:53 -0500
From: Dr. Ken
Subject: Re: Geometry

Hello there!

This is one of the nicer problems I've seen on the SAT (I'm ordinarily 
not a fan of their math problems, or even their answers; they have 
been known to use incorrect answers as correct).  Here's a hint on 
how you might approach it.  You know that the area of the square must 
be 9, right?  So what we want to do is subtract the football-shaped 
region in the middle.

So how do we find its area?  Well, here's what I'd do.  You can find the
area of the part of the square that makes up 1/4 of one of the circles,
right?  If you add two of these regions together, you will get the area 
of the square plus some overlap.  This overlap is the area of the 
football.  And once you've found the area of the football, you can 
subtract it from 9 to find out the answer to the problem.

Thanks for the question, and let us know if you need some more help!

-Ken "Dr." Math
    
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

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