perimeter of the triangle = 24

so side =8

area = 1/2 * b *h = 1/2 *8 * 4sqrt3 = 16sqrt3

also area = product of sides/2D (where D is the diameter of circle)

= 8 ^3 /2D

so 16sqrt3 = 8^3/2D

D = 16/sqrt3

= 9 approx

Thanks Samir for the explanation but ans is 11 (C)
It might be 16/sqrt 2 (Just guessing )

shri_ankur wrote:An equilateral triangle is inscribed on a circle.
Perimeter of triangle is 24 cm.
What is the diameter of the circle?

a) 9 cm
b) 10 cm
c) 11 cm
d) 12 cm
e) 13 cm

Samir is right. The answer is approx 9cm.

The way ive worked it out is the height of the triangle is
√3/2*(side) = 4√3

since radius of circle is 2/3 of height, the diameter is 4/3 of height.

Therefore 4/3*4√3 = 16/√3 = approx 9

Samir, I have a question! According to ur explanation -

perimeter of the triangle = 24

so side =8

area = 1/2 * b *h = 1/2 *8 * 4sqrt3 = 16sqrt3

also area = product of sides/2D (where D is the diameter of circle)

Is the bold lettered statement a hypothesis? I am asking because obviously I don;t know abt that. Can u pls elaborate it little more?

Yes 9 is the correct Ans.
Thanks a lot for the explanation.

Hi All,
Well Amitava

The area of a triangle(any triangle i.e equilateral, isoceles etc) inscribed within a circle is given by the formula

A = a*b*c/4R

(u are free to browse the net if u have doubts on this one, just make sure ur keywords are "Area of a triangle inscribed in a Circle")

where a,b,c are sides of the triangle & R is the circumradius i.e radius of the circle inscribing it.

I only replaced 4R by 2D

hope this is clear.

Ok first of all, you're approach is right but one assumption is wrong. A 3-4-5 sided right angled triangle cannot be a 30-60-90 triangle.

If the side opposite 60 degree is 4, then the side opposite 30 must be 4/√3 and the hypotenuse must be 8/√3. So if the radius is 8/√3, the diameter will be 16/√3, which is approx 9.

Hope this clears your doubt.

the radius of a circle circumscribing a equilateral triangle is given by a/root(3) , where a is the side of the equilateral triangle ..

Gabriel,

Is this a known formula and does it work universally? Or it is more a conclusion.

thanks.

Wongee! U said -

Is this a known formula and does it work universally? Or it is more a conclusion.

Nop it is not a conclusion but a known formula. I have checked it in net. AND above all, thanks to everyone (involved in this discussion) - specially Ratin, Samir and Gabriel - for clearifying my doubts. I really did not know this fact/ theorem/ hypothesis before!

wongee wrote:Gabriel,

Is this a known formula and does it work universally? Or it is more a conclusion.

thanks.

well, i dont know wether this is known or not ( apparently it is not that well known, coz u wouldnt be asking this question in the first place if it was well known ) , but it is a formula that helps fasten things up when u have such questions ... btw this works only for equilateral triangles..

Thanks Gabriel,

Well it coudl be known, and I could be the ignorant one! But, i realize that this can work with an Eq tri only.