Prove that the area of a regular polygon can be calculated by the formula S = 2√3r ^ 2

Prove that the area of a regular polygon can be calculated by the formula S = 2√3r ^ 2 where r is the radius of the inscribed circle.

1. Point O – the center of a circle inscribed in a hexagon. Let’s connect it with two vertices

(A and B). We get an equilateral triangle AOB. AB = r.

2. From point O we draw the height OE.

3. Consider a triangle OAE. The triangle is rectangular, since ∠AEO = 90. °

4. AO² = AE² + OE² (by the Pythagorean theorem).

AE = AB / 2 = AO / 2.

AO² = AO² / 4 + r².

3AO² / 4 = r².

AO² = 4r² / 3.

AO = 2 r / √3. AB = 2 r / √3.

5. We calculate the area of ​​the triangle AOB:

S = AB x OE / 2 = 2 r² / √3.

6. Calculate the area of ​​the hexagon, considering that it is 6 times larger than the area

triangle AOB:

S = 6 х 2 r² / √3 = 2 r²√3, as required.

Answer: S = 2 r²√3.