Total Areas of Alternating Subtriangles in a Regular Polygon with 2n Sides

number of sides

6

8

10

12

S 1	≈	0.15	S 3	≈	0.45	S 5	≈	0.70
S 2	≈	0.17	S 4	≈	0.72	S 6	≈	0.41

S 1	+	S 3	+	S 5	≈	1.30
S 2	+	S 4	+	S 6	≈	1.30

Let P be a point connected to and inside the vertices of a polygon with

2n

sides. Number the triangles counterclockwise from

1

to

2n

. Then the sum of the areas of the even-numbered triangles is equal to the sum of the areas of the odd-numbered triangles.

Drag the point P to change the figure.