Question

A circle has a radius of 6 in. The inscribed equilateral triangle will have an area of: ?

Answer 1

i believe it would be 56.52 (not 100% positive though) because if the radius is 6, then the circumference would be 3.14(6^2)= 3.14(36) which is 113.04 then you would put that over 360 and set up a proportion:
113.04/360 = x/180 (simplified to 20347.2/360) which gives you a final answer of 56.52
hope this helps!

Answer 2

The height of the equilateral triangle will be a median in that triangle, and the centroid will be the centre of the circle.
That centroid splits the median in the ratio of 2 : 1
so the whole median, or the height of the triangle will be 9 in.
Half of the triangle will be a 30-60-90 triangle whose sides are in the ration of 1 : √3 : 2

9/√3 = b/1
so b = 9/√3 or the base is 18/√3

area of triangle = (1/2)(18/√3)(9) = 81/√3 or 27√3 or 46.765

Just thought of an easier way.

From the centre draw the 3 radii, giving us 3 congruent triangle with sides 6 and 6 and contained angle of 120°
Area of one of these = (1/2)(6)(6)sin120°
= 18(√3/2) = 9√3
but we have 3 of them, so the total equilateral triangle is 27√3