Question

a circle is inscribed in an equilateral triangle. Which of the following is nearest the percent of the triangle's area occupied by the circle?
A. 40%
B. 50%
C. 60%
D. 70%
E. 80%

Answer 1

We inscribe the circle inside the equilateral triangle.
Geometrically, this means the center of the circle coincides with the intersection point between all of the angle bisectors of the triangle
So that means, we may make that right triangle.
Where the vertical leg is a radius of the circle.

So then we apply right triangle trig in order to get the radius r in terms of the base b of the triangle tan(30 degree) = Sqrt[3]/3 and that must equal opposite/adjacent which is radius/(base/2) = r/(b/2)
As a result we see that r = Sqrt[3]/6 * b
and so r^2 = b^2/12

(Alternatively, you can use the Pythagorean theorem. Since the hypotenuse is actually a double radius.)

Prior to all this, we can express the area of the equilateral triangle entirely in terms of the base "b" since we know for an equilateral triangle, h = Sqrt[3]/2 * b

Therefore, A1 (of the triangle) = Sqrt[3]/4 * b^2
And A2 (of the circle) = pi r^2 = pi * b^2/12
Divide A2 by A1

So the b^2's divide out.
And we are left with pi/12 * 4 / Sqrt[3]

And we are left with pi/12 * 4 / Sqrt[3]
= pi / (3 * Sqrt[3])
~ 60%

Answer 2

thaank u @miracrown :) tat was indeed a vry clear explanation