# Frustum of a Prism

__Frustum__

When a plane section is taken of a right prism parallel to its end (i.e., perpendicular to its axis), the section is known as a cross-section of the prism, and the two positions of the prism are still prisms. If, however, the plane section taken is not parallel to the ends, the portion of the prism between the plane section and the base is called a frustum.

__The Volume of a Frustum of a Prism__

Figure represents a frustum of a prism whose cutting plane is inclined at an angle to the horizontal. In this case, the frustum can be taken as a prism with base and height .

(i) Volume of the frustum

**ABEF** is a trapezium whose area is

volume of the frustum

i.e.

__Lateral Surface Area of a Prism__

If the cutting plane is inclined at an angle to the horizontal, then from figure we have

or

or

or

Hence

Total surface area = area of the base + area of the section + lateral surface area

**Note:** The lateral surface area of the frustum is the combination of the rectangle and trapeziums whose area can be calculated separately.

__Example__**:**

A hexagonal right prism whose base is inscribed in a circle of radius 2m, is cut by a plane inclined at an angle of to the horizontal. Find the volume of the frustum and the area of the section when the heights of the frustum are 8m and 6m, respectively.

__Solution__**:**

Area of cross-section

Here

area of base

volume of frustum

Area of the section