Math Forum - Project of the Month


November 1998 Honorable Mentions

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Steve Holmes
Grade 10, Germantown Academy, Fort Washington, Pennsylvania

The sum of the angles formed at the tip of the first 5 tip star is 180 degrees. I found this out by using the Sketchpad at school. The sum of the angles formed at the tip of the second 6 tip star is 360 degrees. I used the sketchpad to find this out also. I then formed a seven tip star and the measures of all the angles at the tips added up together were 540 degrees.

I derived the following formula to find the sum of the angles formed at the tips of an "n" pointed star.

(n times 180) - 720

I use n times 180 to find the sum of all the angles of the triangles at the tips. the base angles of Each of these triangles contains 2 exterior angles from the interior polygon. I know that the sum of the exterior angles of a polygon equals 360. Since I took 2 exterior angles at each vertex i multiplied 360 by 2 and thats how i came up with 720. I needed to subtract the 720 degrees from the sum of all the triangles' angles to get the sum of the tip angles from each triangle.

Jenny Kaplan
Grade 8, Castilleja School, Palo Alto, California

I started with the five point star. I labeled all the inner angles in the pentagon a, b, c, d and e. I then looked at the lower base angles of the five triangles surrounding the pentagon. I labeled these angles x, y, z, p and q. There were two of each of these angles. To find the sum of the top angles of this star I made this equation:

Sum = 180*5 - (2x + 2y + 2z +2p + 2q) or
Sum = 900 - 2(x + y + z + p + q)
Since each of the x, y, z, p and q is 180 - a, b, c, d or e, I made the equation:
Sum = 900 - 2(900 - (a + b + c + d + e))
I knew that the sum of the interior angles of a pentagon is 180*(5 - 2) = 540. This made my equation:
Sum = 900 - 2(900 - 540) = 180.
For the second star, I did the same thing and I came up with:
Sum = 180*6 - 2(180*6 - 180*4) = 360.
I then came up with the general case:
Sum = 180*n - 2(180*n - 180(n - 2))
Sum = 180*n - 720

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