Trig question with Hexagon and Circle (area)?

Question

Q> Find the area of a regular hexagon circumscribed about a circle of radius 12 inches (Hint: start by finding the distance from a vertex of a the hexagon to the center of the circle.)

This is the last question for my weekends homework, unfortunately, i was never that good in geometry related works :D

Thanks if you can help.

Answer 1

If you draw the diagram you will find that the regular hexagon is made up of 6 equilateral triangles all circumscribed by the circle and having side length equal to circle's radius.

So you have circle radius 12 inches

1 triangle has side length 12 inches

perpendicular height h can be found by
6^2 + h^2 = 144
h=SQRT(144-36)
= SQRT(108)
= SQRT(2.2.3.3.3)
= 6 x SQRT(3)

And hence area of 1 triangle is 1/2 x base x height
= 1/2 x 12 x 6 x SQRT(3)
= 36 x SQRT(3)

Area of 6 triangles = the hexagon
= 216 SQRT(3)

Answer 2

First answer is correct, of course, but do you know how to get the area of each equilateral triangle? If you draw an altitude, it bisects a side, so you have a rightangled triangle with hypotenuse 12, one side 6, and so the other side is
√(12^2 - 6^2)
=√108 = 6√3

Hence area
= (1/2)*12*6√3
= 36√3
There are six of these, and so total area of hexagon
= 216√3

The general formula for the area of an equilateral triangle with side a is
(a^2)(√3)/4 not all those parentheses are necessary, but best to be certain.

OK, I didn't know first answerer was still working on it -- I've wasted my time!