Posted by **Kaley** on Monday, May 17, 2010 at 4:54pm.

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

- Geometry -
**lily**, Monday, May 17, 2010 at 5:37pm
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!

- Geometry -
**Reiny**, Monday, May 17, 2010 at 6:34pm
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

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