Square Inscribed in a Circle

Date: 09/28/97 at 14:37:01
From: Steve
Subject: 9th grade Math.....Brain Twister

A square with 10-inch sides is inscribed in a circle. What percent of 
the circle is contained within the square?

Date: 10/22/97 at 11:14:04
From: Doctor Mandel
Subject: Re: 9th grade Math.....Brain Twister

Dear Steve,

One way to start this problem is to consider the diameter of the 
circle. You can do this by drawing a line from one corner of the 
square to the opposite corner. Since the square is inscribed in the 
circle this diagonal is the diameter of the circle.  

The diagonal divides the square into two right triangles, so you can 
use the Pythagorean theorem to find the length of the diagonal and 
therefore the diameter of the circle. 

Once you know the diameter of the circle you can find its area with 
the formula "area = pi times the radius squared." The radius is equal 
to half the diameter, and radius squared means the radius multiplied 
by itself.  

Then to find out how much of the circle is covered by the square, 
compare the areas of the two in a ratio. The ratio of the areas is a 
fraction made up of the square's area in the numerator (on the top of 
the fraction bar) and the circle's area in the denominator (under the 
fraction bar). If you multiply this fraction by 100 you get the 
percentage of the circle that is covered by the square.  

For example, 1/2 is .5, multiplied by 100 gives you 50, so 1/2 is 
the same as 50 percent.  Saying it another way 1 is 50 percent of 2.  
You can put the areas of the square and circle into a similar ratio.  
That ratio would be Area(square)/Area(circle).

-Doctor Mandel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/