# Frustum

A frustum is a chopped-off or truncated cone or a pyramid.

It is the 3-dimensional solid shape formed by cutting a cone or a pyramid from the top with a plane parallel to its base.

## Types

Frustum can be of 2 types, depending on the shape from where it is obtained:

1. Frustum of a Cone
2. Frustum of a Pyramid

Popcorn containers, coffee cups, buckets, and lampshades are some common examples of frustums.

Like all other solid shapes, we can calculate the volume and surface area of a frustum.

## Formulas

### Volume

The general formula is:

Let us solve an example involving the above concept.

Calculate the volume of a frustum with base areas of 81 cm2, and 121 cm2, and a height of 12 cm.

Solution:

As we know,
Volume (V) =${\dfrac{1}{3}h\left( B_{1}+B_{2}+\sqrt{B_{1}B_{2}}\right)}$, here B1 = 81 cm2, B2 = 121 cm2, h = 12 cm
${\therefore V=\dfrac{1}{3}\times 12\left( 81+121+\sqrt{81\times 121}\right)}$
= 1204 cm3

### Surface Area

The general formula to calculate the surface area for both frustum of a cone and a pyramid is:

Lateral Surface Area (LSA) or curved surface area is the area of only the curved surface. The formula is:

Lateral Surface Area (LSA) = ${\dfrac{1}{2}\left( P_{1}+P_{2}\right)\times l}$

Find the surface area of a frustum with base areas of 64 cm2, and 144 cm2, base perimeters of 32 cm and 48 cm, and a slant height of 14 cm.

Solution:

As we know,
Surface Area (SA) = ${\dfrac{1}{2}\left( P_{1}+P_{2}\right)\times l+B_{1}+B_{2}}$, here P1 = 32 cm, P2 = 48 cm, B1 = 64 cm2, B2 = 144 cm2, l = 14 cm
${\therefore SA=\dfrac{1}{2}\left(32+48\right)\times 14+64+144}$
= 768 cm2

Find the lateral area of a frustum with base perimeters of 20 cm and 28 cm, and a slant height of 9 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\dfrac{1}{2}\left( P_{1}+P_{2}\right)\times l}$, here P1 = 20 cm, P2 = 28 cm, l = 9 cm
${\therefore LSA=\dfrac{1}{2}\left( 20+28\right)\times 9}$
= 216 cm2

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