A frustum is formed when you cut a cone or pyramid with a plane that is
parallel to the base and remove the top portion. The resulting solid has two
parallel bases: a larger “bottom” base with area \(A_1\) and a smaller “top” base with area \(A_2\).
The key length for volume is the perpendicular height \(h\), meaning the shortest distance between the two base planes.
A powerful fact is that the volume of a frustum depends only on \(A_1\), \(A_2\), and \(h\), regardless of whether the frustum is right or oblique:
\[
V=\frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1A_2}\right).
\]
This formula comes from similarity: the removed small cone/pyramid is similar to the original, so areas scale by the square of a linear ratio and volumes scale by the cube.
Subtracting the “small” volume from the “big” volume yields the expression above. Because similarity uses parallel slices, the formula works for cone frustums and regular pyramid frustums.
Cone frustum (right case)
With bottom radius \(R\) and top radius \(r\), the base areas are \(A_1=\pi R^2\) and \(A_2=\pi r^2\).
The right frustum has a single slant height (generator length)
\[
s=\sqrt{h^2+(R-r)^2}.
\]
Its lateral surface area is
\[
A_L=\pi(R+r)s,
\]
and the total surface area adds the two circular bases:
\[
A_T=A_L+\pi R^2+\pi r^2.
\]
Regular pyramid frustum (right case)
For a regular pyramid frustum, each side face is a congruent isosceles trapezoid. If \(P_1\) and \(P_2\) are the perimeters of the bottom and top bases,
and \(l\) is the slant height measured along a face (perpendicular to the parallel edges of the trapezoid), then
\[
A_L \approx \frac{P_1+P_2}{2}\,l,\qquad A_T=A_L + A_1 + A_2.
\]
For a right regular pyramid frustum, \(l\) can be computed from the difference of base apothem values (center-to-side distances).
Inclined/oblique note: volume always uses the perpendicular height \(h\). Lateral area depends on slanted geometry; in an oblique frustum, slant heights may vary by direction,
so a single input slant height is typically an effective/average value.
