In this note, I will prove that the area formed by connecting every kth vertex of a regular polygon is when and n=number of sides of the original polygon and a represents the side length of the original polygon.
Can someone link me to a website so I can read up on it? I haven't been able to find one.
Step one: determining the number of sides
This step is rather simple. The number of sides of the original polygon R divided by k will yield the new polygon’s number of sides.
This is because #vertices=#edges. And since we’re dividing the number of vertices by k, the number of edges will be divided by k as well.
Therefore, the number of sides
Step two: determining the length of each side
Ok, this is the hard part. For this part, I will refer to the second picture (also, NOTE 1,2,3,4 represent possible values of k). Begin by looking at the angled formed by . It’s quite obvious that each angle is congruent. Also, . Now, if we’re going to connect every kth vertex starting with vertex J, the angle (NOTE: point k varies in position) formed will be equal to .
Now, the one thing that remains constant no matter what vertex we chose is the length from the center of the original polygon to each vertex, AKA: the radius. The radius r is the hypotenuse of a right triangle with one leg as its apothem and the other leg being . The angle at P of the right triangle is . Therefore, .
Now, using Law of Co-sines, we can find M (length of segment ) in terms of and r (r is in terms of a and here).
Finding the area
This step isn't so hard, in this note, I prove that the area of any regular polygon can be represented as . (NOTE: that I substituted a=m from the other note and the number of sides is rather than n because in this note I’m using different variables than in the other note)
Plugging in our value of yeilds