regarding math...

Question

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

Needs to be in radicle form



Answer 1

The base of the equilateral triangle is 2*3√3 because the triangle in red above is a 30-60-90 special right triangle with a hypotenuse of 6 making the bottom 3√3. The height of the equilateral triangle is 9 because the length of the short side of that same 30-60-90 special right triangle is 3 and we add that to the radius of 6 to get the total height. The area = 1/2*b*h = 1/2(6√3)(9) = 27√3

Answer 2

I couldn't think of a particularly easy way to get this geometrically.

Sooooo...

The equation of a circle with radius with center at (0,6) is x² + (y² - 36) = 36

The equation of the side of the inscribed equilateral triangle is y = 12 - x√3 (because the slope is the -tan 60°).

Those two equations intersect at (3√3,3)

The area of the triangle is, therefore, (12-3)*√3

If I think of a better way to do it, I will get back to you.