How to derive the formula for voulume of frustum of a cone?

- How to derive the relation bnetween the orginal cone radius / height to the portion of the cone that is removed?

The volume of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex:

where *B*_{1} is the area of one base, *B*_{2} is the area of the other base, and *h*_{1}, *h*_{2} are the perpendicular heights from the apex to the planes of the two bases.

Considering that

the volume can also be expressed as the product of the height *h* = *h*_{2}−*h*_{1} of the frustum, and the Heronian mean of their areas:

Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary no, the square root of negative one.

In particular, the volume of a circular cone frustum is

where *π* is 3.14159265..., and *R*_{1}, *R*_{2} are the radii of the two bases.

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

where *a*_{1} and *a*_{2} are the sides of the two bases.

Surface area

The surface area of a right circular cone frustum is

where *R*_{1} and *R*_{2} are the base and top radii respectively.

The surface area of a right frustum whose bases are similar regular *n*-sided polygon is

where *a*_{1} and *a*_{2} are the sides of the two bases.