> the lateral surface area of a regular hexagonal > pyramid is 288 cm squared. if the slant height of > each face is 12 cm, find the volume of the pyramid.
A hexagonal pyramid has, of course, six lateral faces so each has area 288/6= 48 cm^2. The area of a triangle is "one half height times base" so the base length, and so the length of each side of the base hexagon, is 2(48)/12= 2(4)= 8 cm. You can subdivide the hexagon into 12 right triangles having one leg of length 4 cm and opposite angle of 360/12= 30 degrees. That means that the distance from the center of the base to the center of a side is 8 cm and, by the Pythagorean theorem, the distance from the center of the base to a vertex is 4 sqrt(3) cm.
With that information there are a number of ways to find the volume, ranging from looking up a formula to using Calculus.