Difficult Math Question #42 - Geometry

[800guy]

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

[ajith]

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???

[rajs.kumar]

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

[ajith]

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

I disagree with you here. Is it not 2/3 of the height?

[limits660]

I tired to figure this one and cant

beyond my current ability

[rajs.kumar]

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)

[800guy]

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)

[beeparoo]

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