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# What is the ratio of the area of a circle to the area of an

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Manager
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What is the ratio of the area of a circle to the area of an [#permalink]  20 Mar 2008, 06:08
What is the ratio of the area of a circle to the area of an equilateral triangle inscribed in the circle?
Pi*sqr/4
3/2
Pi*/2
3*Pi/5
4*sqr 3/9
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Re: What is the ratio of the area [#permalink]  20 Mar 2008, 17:26
As triangle is equilateral so all its internal angle are 60.
3 radii extending from centre of the circle to the verticies A, B, and C will equally divide 60 so resulting angle will be 30.

Now Height of triangle will be r+h.
Where (s/2)/r = cos30 and h/r = sin30
From above we have s=r*sqrt(3) and h=r/2
So Area of triangle is = 1/2 * s * (r+h) = 1/2 * r*sqrt(3) * 3r/2 = 3*r^2*sqrt(3)/4

Area of Circle = pi*r^2

Ratio = pi*r^2/3*r^2*sqrt(3)/4 = 4*pi*sqrt(3)/9

I think it closely resembels Answer E. (Might be pi is accendently omitted there).
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Re: What is the ratio of the area [#permalink]  20 Mar 2008, 18:00
This is how I visualise the circle and the inscribed equilateral triangle ABC to look like.

Since an equilateral triangle has equal sides, the angle ABC, ACB and CAB = 60deg. O is center of the circle. The radius of the circle forms two legs, cutting the equilateral triangle ABC into 3 smaller triangles.

I started with triangle AOE. Angle OAE = 30deg (60deg / 2), angle AOE = 60deg. Thus, line AO = sqrt 3, line OE = 1, line EA = 2.

Area of triangle ABC = 1/2 * 1 * 2 * 6 = 6 ( * 6 being small triangle AOE, OEB, BOF, FOC, COD, AOD)

Area of circle = pi * r^2 = pi * (sqrt 3) ^ 2 = 3 * pi (where radius = line AO = sqrt 3)

Therefore, ratio of area of circle to area of triangle
= 3pi / 6
= pi / 2

I hope my working is not too confusing
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Jimmy Low, Frankfurt, Germany
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Re: What is the ratio of the area   [#permalink] 20 Mar 2008, 18:00
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