Find the area of a regular hexagon inscribed in a circle of a radius 12 inches.?

regular-hexagon

#1

FIND THE AREA OF A REGULAR HEXAGON INSCRIBED IN A CIRCLE OF A RADIUS 12 INCHES.?

Answer:

First, as a QC step, you should expect the answer to be a little less than pir^2 = pi*12^2 = 144 pi = 452.3893
The hexagon can be separated into 6 triangles with each triangle having 1 side the hexagon side and the other 2 sides from the center to the 2 ends of the hexagon side.
With a hexagon, the central angle is 360/6 = 60 degrees, so the other 2 angles are also 60 degrees. Thus, we have an equilateral triangle with 3 sides of length 12. You can calculate this area by seeing there is an altitude from a vertex to the midpoint of the other side; thus, as the altitude is part of a right triangle with lengths 6 and 12, it has length 6 * sqrt(3)
Thus, A = 1/2 bh = 1/2 * 6 sqrt(3) * 12 = 36 sqrt(3)
Then, the hexagon area = 6 * 36 * sqrt(3) = 216 * sqrt(3) =
374.122974434877
Note that this is somewhat less than 452.3893, the circle with the same radius