Question

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

3-inch radius

A = sq. in.

Answer 1

The 3-inch radius measurement they gave you is the radius of the circumscribed circle around the triangle. The good news is that there's an easy way to find the height of an equilateral triangle given this radius.

The radius of the circumscribed circle of an equilateral trianlge is 2/3 the height. So, 3=(2/3)h. Solving for h, we get the height to be 9/2.

We still need one more piece, though (unless you have learned trigonometric functions). We need the length of the base of the triangle. To find the base of an equilateral triangle given it's height, we multiply the height by \[2/\sqrt{3}\]So,\[(9/2)(2/\sqrt{3})=9/\sqrt{3}=3\sqrt{3}\]We can now use the A forumla for a triangle.
\[A=(1/2)bh=(1/2)(3\sqrt{3})(9/2)=27\sqrt{3}/4\]