The Star Medallion

Geometry Level 3

A pentagram is inscribed inside a circle with radius r r as shown in the picture above. The formula that puts the area of the pentagram S S in terms of the circle's radius r r can be expressed as follows:

S = a 10 a − b a c r 2 , \large S = \dfrac{a \sqrt{10a-b\sqrt a}}c r^2 \; ,

where a , b a,b and c c are positive integers with a a square-free and a , c a,c coprime.

Find the value of a + b + c a+b+c .


The answer is 31.

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Ruben Calzadilla-Badra by Ruben Calzadilla-Badra , 22, Canada

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1 solution

Ahmad Saad
Apr 10, 2016

Area of pentagram is equal to 10 times the area of triangle O A B OAB .

a sin ⁡ 3 6 ∘ = r sin ⁡ 12 6 ∘ = r sin ⁡ 5 4 ∘ = r cos ⁡ 3 6 ∘ \dfrac{ a}{\sin36^\circ} = \dfrac r{\sin126^\circ} = \dfrac r{\sin 54^\circ} = \dfrac r{\cos36^\circ}

Because a = r ⋅ tan ⁡ 3 6 ∘ a = r \cdot \tan 36^\circ , the area of triangle O A B OAB is equal to 1 2 r ⋅ a ⋅ sin ⁡ 1 8 ∘ = 1 2 r 2 tan ⁡ 3 6 ∘ sin ⁡ 1 8 ∘ \dfrac12 r \cdot a \cdot \sin18^\circ = \dfrac12 r^2 \tan36^\circ \sin18^\circ .

Multiply this number by 10, and we get the required area of 5 4 10 ⋅ 5 − 22 5 \dfrac54 \sqrt{10\cdot 5 - 22\sqrt5} .

Note : It can be shown that we can evaluate the following trigonometric expressions

tan ⁡ 3 6 ∘ = 5 − 2 5 , sin ⁡ 1 8 ∘ = 1 4 ( 5 − 1 ) . \tan 36^\circ = \sqrt{5 - 2\sqrt5} , \quad \sin18^\circ = \dfrac14 (\sqrt5 - 1) \; .

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In the question, the only variable given is "r" whereas the answer is in terms of 2 variables. So the question is incomplete.

About the solution given: How can one assume that "a" is the length of the arm?

Shashankh , C - 3 years, 5 months ago

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@Ahmad Saad Please do your best to avoid letting notation do double duty.

@Shashankh , C The idea is that "a" is used for the arm length in the solution, in order to calculate the final form of the answer. Subsequantly, we then see that "a" in the final form is equal to 5. It is not ideal to let "a" represent different things, as it leads to confusion like yours.

Calvin Lin Staff - 3 years, 5 months ago

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I'm sorry for misunderstanding that part. My actual doubt was how could the measure of angle OAB be equal to 18°? It depends on the length OB which is a variable independent of "r" and could range from 0 to r(and obviously the area depends upon it). That's what I meant when I said that the answer would be in terms of 2 variables- "r", and the angle OAB, which is directly related to the length OB.

Shashankh , C - 3 years, 5 months ago

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@Shashankh , C – Ah. I've clarified that we have a pentagram, instead of just a 5-pointed star.

Calvin Lin Staff - 3 years, 5 months ago

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