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.
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) orSince each of the x, y, z, p and q is 180 - a, b, c, d or e, I made the equation:
Sum = 900 - 2(x + y + z + p + q)
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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