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Square Inscribed in a Circle

```
Date: 09/28/97 at 14:37:01
From: Steve

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
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/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Triangles and Other Polygons

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