Answer:
Total area = 6(16√3) = 96√3
Step-by-step explanation:
Important that you understand the meaning of "apothem," looking that up on the 'Net if necessary. A regular hexagon can be broken down into 6 identical, pie-slice-shaped figures; we find the area of the hexagon by finding the area of one such slice and multiplying the result by 2.
An apothem is a line from the center of the circle to the edge of the slice, perpendicular to that edge, representing the height of the slice as measured from that edge to the center of the circle. There are 6 such slices. The apothem is also a measure of the height of such a triangular slice. If the radius is 8, then half the base of the slice is found using the Pythagorean Theorem: (b/2)² + (4√3)² = 8².
Then (b²/4) + 16(3) = 64, or
b²/4 = 64 - 48, or b²/4 = 16, which yields b² = 64, or b = 8.
Then the area of one slice is A = (1/2)*(base)(height) = (1/2)(8)(4√3), or
A = 16√3. Multiply this result by 6 to find the area of the entire hexagon:
Total area = 6(16√3) = 96√3
