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Difficult Math Question #42 - Geometry

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Re: Difficult Math Question #42 - Geometry

by ajith » Mon Oct 30, 2006 10:27 pm
800guy wrote:The area of an equilateral triangle is 9.what is the area of it circumcircle.

A.10PI B.12PI C.14PI D.16PI E.18PI
area of an equ. triangle = 3^1/2/4 *a^2 ( where a is the side)
=9
=> a^2 = 36 / 3^(1/2)
=> a = 6/ 3 ^ (1/4)
radius of the cirumcircle r =a/3^(1/2)
= 6/ 3^(3/4)
area = PI* r^2= PI * (6/ 3^(3/4))^2
= PI * (36/3^3/2)
= PI * 12/3^1/2

Which will be less than 10 PI anyway

So I dont think I have an answer here, are you sure that the area of the triangle is 9???
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by rajs.kumar » Tue Oct 31, 2006 2:24 am
Answer is D.

0.5 x b x h = 9

b = base of the triangle
h = height of the triangle

In a equilateral triangle base and height are related in the following manner, b = h/sqrt(3)

h^2 = 18 x sqrt(3)

Also for a circumcircle diameter d = height of the equilateral triangle.

=> Area of circle = (PI x d ^ 2)/2 ~= 16PI
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by ajith » Tue Oct 31, 2006 4:43 am
rajs.kumar wrote:
Also for a circumcircle diameter d = height of the equilateral triangle.

I disagree with you here. Is it not 2/3 of the height?
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by limits660 » Tue Oct 31, 2006 5:38 am
I tired to figure this one and cant

beyond my current ability
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by rajs.kumar » Tue Oct 31, 2006 8:35 am
ajith wrote:
rajs.kumar wrote:
Also for a circumcircle diameter d = height of the equilateral triangle.

I disagree with you here. Is it not 2/3 of the height?
Yes I am wrong and you are correct mate. I have to be more careful when posting, it is my mistake.

If we assume the side of the triangle to be b and height to be h they are related in this way

sin 60 = h/b => h = b x sqrt(3) / 2 --- (1)

area of triangle = 9 => b x h = 18 --- (2)

(1) in (2) => b^2 = 36/sqrt(3) --- (3)

the radius of the circumcircle r is related to the side b of the triangle in the following manner.

r/b/2 = sec 60 => r = b/sqrt(3) ---- (4)

using (4) So area of the circumcircle is PI x r^2 = PI x b^2/3 --- (5)

(3) in (5) => area = PI x 12/sqrt(3)

which is the same as what ajith got.

If you can remember the formula directly you will get the same result. The formula for area of the circucircle of equilateral triangle is 1/3 x PI x b^2 where b is the length of the side.

The formula gives the same value, area = PI x 12/sqrt(3)
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OA

by 800guy » Wed Nov 01, 2006 10:58 am
OA:

area = sqrt(3) / 4 * (side)^2 = 9

so , ( side )^2 = ( 9*4 ) / sqrt(3)

Height = H = sqrt(3)/2 * (side) , so H^2 = 3/4 * Side^2 = 3/4 * (9*4) / sqrt(3)

Radius of CircumCircle = R = 2/3 of ( Height of the Equilateral Triangle )

so , area of Circumcircle = PI * R^2

=> PI * 4/9 * H^2

=> PI * 4/9 * 3/4 * (9*4) / sqrt(3)

=> PI * 4 * sqrt(3)
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by beeparoo » Sat Aug 25, 2007 8:49 am
Why is the final answer not even a choice in the original selection of answer choices?

That is, 4*sqrt(3)*PI doesn't match A, B, C, D, or E
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