Question

Find the area of an equilateral triangle (regular 3-gon) with the given measurement.

3-inch radius

A = sq. in.

Answer 1

In an equilateral triangle, the incenter, circumcenter, orthcenter, and centroid meet at the same point.The distance from this point to 2 vertices is the radius of the triangle. If we draw, say 3 medians, this 3 segment will divide the triangle into 6 30-60-90 right triangles where the radius id the hypotenuse. The longer leg of the 30-60-90 is 1/2 of the side of the equilateral triangle. Thus
s = 2(3 sqrt (3)/2) = 3 sqrt (3)
The area of the equilateral triangle is
A = s^2 (sqrt (3))/4 ={ [3 sqrt (3)]^2 * sqrt (3)} /4 = (27 sqrt (3))/4