Big Pentagon Small Pentagon

Geometry Level pending

The vertices of the small pentagon (purple) are the midpoints of the sides of the big pentagon (green). Find the ratio of the area of the small pentagon to the area of the big pentagon.

In the answer options, φ = 1 + 5 2 \varphi = \dfrac{1 + \sqrt{5}}{2} denotes the golden ratio .

φ + 1 4 \dfrac{\varphi +1}{4} 1 φ + 1 \dfrac{1}{\varphi + 1 } None of the other answers 1 φ \dfrac{1}{\varphi} φ − 1 \varphi - 1

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Chew-Seong Cheong
Nov 15, 2020

Let the radius of the big pentagon be 1 1 . Then the radius of the small pentagon r s = cos ⁡ 3 6 ∘ = 1 + 5 4 = φ 2 r_s = \cos 36^\circ = \dfrac {1+\sqrt 5}4 = \dfrac \varphi 2 and the ratio of their areas A s A B = r s 2 1 2 = φ 2 4 = φ + 1 4 \dfrac {A_s}{A_B} = \dfrac {r_s^2}{1^2} = \dfrac {\varphi^2}4 = \boxed{\dfrac {\varphi +1}4} .

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