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# An equilateral triangle is inscribed in a circle. How many times great

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Math Expert
Joined: 02 Sep 2009
Posts: 58445
An equilateral triangle is inscribed in a circle. How many times great  [#permalink]

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31 Oct 2018, 02:00
00:00

Difficulty:

65% (hard)

Question Stats:

45% (02:06) correct 55% (02:19) wrong based on 37 sessions

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An equilateral triangle is inscribed in a circle. How many times greater is the area of the circle than the area of the triangle?

A. π/√3

B. 3π/4

C. 4π/(3√3)

D. 3

E. 2π/√3

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Re: An equilateral triangle is inscribed in a circle. How many times great  [#permalink]

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31 Oct 2018, 06:56
1
See diagram below, O divides PQ in 2:1 ratio (property of equilateral triangle)
PO = r, PQ = r + r/2 = 3r/2
Area of equilateral triangle = $$\sqrt{3}$$*$$a^2$$/4

Perpendicular bisector of a side has length $$\sqrt{3}$$*a/2 = PQ

Hence, 3r/2 = $$\sqrt{3}$$*a/2 which gives r = a/$$\sqrt{3}$$

Hence area of circle = pi*$$a^2$$/3

Ratio = (pi*$$a^2$$/3)/($$\sqrt{3}$$*$$a^2$$/4) = 4*pi/3$$\sqrt{3}$$

Hence Option C
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Re: An equilateral triangle is inscribed in a circle. How many times great   [#permalink] 31 Oct 2018, 06:56
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