Math Forum - Project of the Month

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November 1998 Winner

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Tommy Hurlbutt
Grade 10, La Cañada High School, La Cañada, California

Question:

  1. Find the sum of the angles formed at the tips of both stars
  2. Find a formula for the sum of the sum of the angle measurements at the tips of an n-pointed star.
Answer:
  1. The sum of the angles formed at the tips of the five pointed star is 180; the sum of the angles formed at the tips of the six pointed star is 360.
  2. The formula for the sum of the angle measurements at the tips of an n-pointed star is f(n)=180(n)-720 where n is an integer greater than 4.
Explantion:

Let's start off with the five pointed star:

If you get confused at any time, just refer back to the top diagram.

Before we jump into the problem, lets get the facts straight:

With that out of the way, we can attact the problem.

Based on this diagram, we can see that we have:

I labeled the diagram in the following way:

Red angles- Set 1 of the exterior angles of Pentagon FGHIJ
Blue Angles- et 2 of the exterior angles of Pentagon FGHIJ
Green Angles- Vertex angles

Lets deal with the five triangles now:

FINALLY, we have our answer: the sum of the vertex angles is 180!!!

If you are still interested, continue on:


For the six pointed the star, we can do ALL of that work again, or we can take a shortcut...

We see that the figure is made up of two triangle, highlighted in the figure. We then can say that the sum of the vertex angles is 180(2), or 360. We CAN do ALL that work to arrive at the same answer, but I think I would be too time consuming.


To find the equation, we need to find the varible:
Number of triangles and the sides of the polygon that form the n-pointed figure.
Even though the number of sides of the polygons changes, the sum of each set of exterior angles doesn't change ( 2 sets of 360)

We now can refer back to our original equation, and insert the constant and varible to get:

(n)measurement of the vertex angle+720=180(n) or sum of vertex angles [or (f)n] =180(n)-720.
If we set up the equation to set an inequality, we get n>4 (greater than because the function needs to be greater than 0) and since we can't have numbers other than postive integers, we can see that f(n)=180(n)-720 where n is an integer greater than 4.
If you were able to stay interested, I thank you for your attention and I hope you understand.

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