What is the volume of a truncated octagonal based pyramid?

Answer 1

The volume of a pyramidal frustum whose bases are n-sided regular polygons is

where a1 and a2 are the sides of the two bases.

Note: cot(x) = 1 / tan(x)

For example, I was not provided my a's to calculate the volume my pyramidal octagonal frustrum. I was provided only an overal width (let overal width = W) so I had to calculate a1 with this equation:

a1 = W * (√2 / (2 + √2))

Then with the given h I solved for the overal width for the top octagon using trigometry;

Used the same a1 equation but for a2;

Then used this equation to find my volume:

It works, I tried it and proved it with AutoCAD using the command MASSPROP

Answer 2

The volume of a truncated octagonal based pyramid can be calculated using the formula V = (1/3)h(A1 + sqrt(A1A2) + A2), where h is the height of the pyramid, A1 is the area of the larger octagonal base, and A2 is the area of the smaller octagonal base. This formula takes into account the three-dimensional shape of the pyramid and the difference in area between the two bases. By plugging in the values for h, A1, and A2, you can calculate the volume of the truncated octagonal based pyramid.