Answer:
750√3 cm³
Explanation:
This is a hexagonal pyramid, meaning that the base is a hexagon of side length 10 cm. The area of this base is 6 times the area of a triangle formed by drawing radii from two adjacent vertices to the center of the hexagon. This is an equilateral triangle with 3 sides of length 10 cm and identical angles (that is, all the angles are 60°).
We need to determine the "height" of one of these triangles. If h is the horiz. distance from the side marked 10 cm in the drawing, then h/(10 cm) = sin 60°, or:
h √3
---------- = ------
10 cm 2
Cross-multiplying results in:
2h = 10√3, and so h = 5√3.
The area of one of the six triangles making up the base of this pyramid is
found using the formula A = (1/2)(base)(height), which here comes out to:
A = (1/2)(10 cm)(5√3 cm) = 25√3 cm²,
and so the total area of the pyramid's base is 6(25√3 cm²) = 150√3 cm².
Going back to the formula for the volume of a pyramid,
A = (1/3)(150√3)(15) cm³ = 750√3 cm³