Problematic n- gon

Geometry Level pending

The ratio between the area of a regular N- sided polygon and area of a square is tan ⁡ 89 45 \frac{\tan 89} {45} . Given that their perimeters are equal, what is N?


The answer is 180.

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1 solution

Unstable Chickoy
Jun 15, 2014

S S = 45 S_S = \sqrt{45}

N S N = 4 S S NS_N = 4S_S

S N = 4 45 N S_N = \frac{4\sqrt{45}}{N}

tan ⁡ 45 = N 2 x 2 sin ⁡ 360 N \tan{45} = \frac{N}{2} x^2 \sin{\frac{360}{N}} where x x is the radius of circumscribing circle

sin ⁡ 360 2 N = S N / 2 x \sin{\frac{360}{2N}} = \frac{S_N / 2}{x}

sin ⁡ 360 2 N = S N 2 x \sin{\frac{360}{2N}} = \frac{S_N}{2x}

Substitute value of S N S_N then solve for x x

x = 2 45 N sin ⁡ 360 2 N x = \frac{2\sqrt{45}}{N\sin{\frac{360}{2N}}}

Plugging the value of x x

tan ⁡ 45 = N 2 [ 2 45 N sin ⁡ 360 2 N ] 2 sin ⁡ 360 N \tan{45} = \frac{N}{2} [\frac{2\sqrt{45}}{N\sin{\frac{360}{2N}}}]^2 \sin{\frac{360}{N}}

N = 180 N = \boxed{180}

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