thats area of triangle /area of circle.

Rt 3/4 side^2/Pi R^2

solving: 63/152
Approx 41.44%

3 . sqrt(3) / 4 . pi

These answers are correct. The fastest way to do this without memorizing this very specific formula is to memorize these less specific formulas:

In a 30-30-120 triangle (as is created by 2 radii and one side of the inscribed equilateral triangle), the ratio of sides is x : x : x*sqrt(3).

And, the other formula you should definitely memorize is the one for area of an equilateral triangle:
https://blog.gmatboost.com/2011/08/02/me ... -triangle/

So, if the radius of the circle is r, then the side of the triangle is r*sqrt(3), and the area of the triangle is:
[r*sqrt(3)]^2 * sqrt(3)/4 = 3r^2 * sqrt(3)/4

The area of the circle is pi*r^2.

So, the fraction taken up by the triangle is
[3r^2 * sqrt(3)/4] / [pi*r^2] -> cancel out the r^2
= [3 * sqrt(3)/4] / [pi]
= [3 * sqrt(3)] / [4pi]

Check out the next challenge question:
https://www.beatthegmat.com/gmat-boost-c ... tml#406020

gmatboost wrote:Here is a GMAT Boost Challenge Question.

An equilateral triangle is inscribed in a circle. What fraction of the circle's area is contained within the triangle?

The question is asking:



Area of the triangle
--------------------------
Area of the circle



If you can't memorize the formula (like me), it would be useful to draw a diagram for such questions.
I have attached a diagram for those who want to take a stab at it...

You will need to know...
30:60:90 rule
and
a few triangle properties
(you will have to memorize these)

Try to solve it now...

@ gmatboost - you mention a useful formula but the explanation isn't clear without a diagram. the manhattan gmat book doesn't talk about 30-30-120 triangle.

@robosc9 - thanks for providing the diagram.