Stars in a Flag

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

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/