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### Stars in a Flag

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Date: 04/15/99 at 15:06:01
From: Eric Hanlon
Subject: Area of a star

The original question was to find the area of the colors in the
American Flag. The last area for me to find is the area of the stars.
The only given is that the sides of the stars are 13mm (total of ten
sides).

My only guess to try to solve this problem is to somehow divide the
triangles of the star into right angles, determine the area, and try
to figure the area of the pentagon in the center. Any help would be
appreciated.

Thanks,
Eric Hanlon
```

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Date: 04/15/99 at 16:30:33
From: Doctor Wilkinson
Subject: Re: Area of a star

Nice problem, Eric, and you have a good idea: you just need to run
with it.

If I understand what you're saying, you have a pentagon and you make a
star in the usual way by drawing the five diagonals. Then the area
you're looking for is the area of the pentagon minus the sum of the
areas of the five triangles that you would have to cut away to make
the star.  Right so far?

Let's start with the five triangles. Each of them can be divided into
two right triangles. You know the hypotenuse of each of them, namely
13mm. If you could figure out an angle you could then find the base,
which is a plus, because that's one of the sides of the pentagon, and
the height, which would give you the area. So see if you can figure
out the angle.

To find the area of the pentagon, it's probably easiest to draw the
line segments from the center of the pentagon to each of the vertices.
Now you have five isosceles triangles whose areas add up to the area
of the pentagon. Again, you can divide each isosceles triangle into
two right triangles, and one side is half the side of the pentagon,
which you've already found out from the first part of the problem. All
you have to do is figure out an angle and you can finish it off
using trigonometry.

Good luck, and get back to us if you get stuck!

- Doctor Wilkinson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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