Mathematics
Sum of internal angles of a 5-point star
Easier than you think
I remember when I was in the 8th grade, our class was given a HW problem. The problem was the following:
Find the sum of the interior angles of a 5-point star.
At first, it seems to be a difficult (almost impossible) to solve this problem. So little information is given.
However, this problem is very easy to solve. All you need is just a good knowledge of geometry.
Solution:
A sum of exterior angles of a polygon is equal to 360°.
Note: this statement is true for all polygons. Since, the sum of interior angles is equal 180°*(n-2), where n is a number of angles, and each exterior angle is supplementary to its interior angle (form 180° angle together), then the sum of exterior angles is equal to 180°*n-180°(n-2)=180°*n-180°*n+360°= 360°.
The sum of “green” interior angles is 360° (similar for the “blue” angles). Therefore, the sum of the colored angles is 720°.
The sum of interior angles of a triangle is 180°. There are 5 triangles, so the sum of all of their interior angles is 5*180°=900°.
Finally, subtract 720° from 900° to find the sum of interion angles of the star.
900°-720°=180°
The sum of the interior angles of a 5-point star is 180°. And it’s true for all 5-point stars.
