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# Circle O is inscribed in equilateral triangle ABC. If the area of ABC

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Math Expert
Joined: 02 Sep 2009
Posts: 61358
Circle O is inscribed in equilateral triangle ABC. If the area of ABC  [#permalink]

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15 Nov 2019, 00:28
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Difficulty:

55% (hard)

Question Stats:

58% (02:25) correct 43% (02:34) wrong based on 40 sessions

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Circle O is inscribed in equilateral triangle ABC. If the area of ABC is $$24 \sqrt{3}$$, what is area of circle O?

A. $$2\pi\sqrt{3}$$

B. $$4\pi$$

C. $$4\pi\sqrt{3}$$

D. $$8\pi$$

E. $$12\pi$$

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Joined: 16 Feb 2015
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Re: Circle O is inscribed in equilateral triangle ABC. If the area of ABC  [#permalink]

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17 Nov 2019, 20:51
2
Bunuel wrote:
Circle O is inscribed in equilateral triangle ABC. If the area of ABC is $$24 \sqrt{3}$$, what is area of circle O?

A. $$2\pi\sqrt{3}$$

B. $$4\pi$$

C. $$4\pi\sqrt{3}$$

D. $$8\pi$$

E. $$12\pi$$

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Area of Equilateral Triangle: (√3/4)*a^2
therefore, (√3/4)*a^2= 24√3; a=4√6

The radius of a circle when inscribed in Equilateral Triangle: r = (√3a )/ 6
r= (√3*4*√6)/6
r= 2√2

Area of circle= pi*r^2
therefore, the area of a circle: pi* (2√2)^2 = 8pi

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Re: Circle O is inscribed in equilateral triangle ABC. If the area of ABC  [#permalink]

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15 Nov 2019, 00:41
side of ∆ = 24√3 = √3*s^2/4
s= 4√6
radius of inscribed circle = s*√3/6
r= 4√6*√3/6
r= 2√18/3
area of circle = pi * ( 2√18/3)^2 = $$8\pi$$
IMO D

Bunuel wrote:
Circle O is inscribed in equilateral triangle ABC. If the area of ABC is $$24 \sqrt{3}$$, what is area of circle O?

A. $$2\pi\sqrt{3}$$

B. $$4\pi$$

C. $$4\pi\sqrt{3}$$

D. $$8\pi$$

E. $$12\pi$$

Are You Up For the Challenge: 700 Level Questions
Intern
Joined: 23 Jan 2020
Posts: 3
Re: Circle O is inscribed in equilateral triangle ABC. If the area of ABC  [#permalink]

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13 Feb 2020, 03:00
Since GMAT doesn't really care how we get the answer, I tried approximation here. For those who did not remember the side to radius relations,
Area for triangle= 24*3^1/3 = 40 approx.
Area of an inscribed circle in an equilateral triangle is roughly 60% of the traingles area = 40*3/5 = 24
Of the given options, 8pi is closest to 24 hence
IMO D
Re: Circle O is inscribed in equilateral triangle ABC. If the area of ABC   [#permalink] 13 Feb 2020, 03:00
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