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/
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