Table of Contents

Last modified on March 28th, 2023

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.

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

**Frustum of a Cone****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.

The general formula is:

Let us solve an example involving the above concept.

**Calculate the volume of a frustum with base areas of 81 cm ^{2}, and 121 cm^{2}, 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 B_{1} = 81 cm^{2}, B_{2} = 121 cm^{2}, h = 12 cm

${\therefore V=\dfrac{1}{3}\times 12\left( 81+121+\sqrt{81\times 121}\right)}$

= 1204 cm^{3}

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 cm ^{2}, and 144 cm^{2}, 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 P_{1 }= 32 cm, P_{2 }= 48 cm, B_{1} = 64 cm^{2}, B_{2} = 144 cm^{2}, l = 14 cm

${\therefore SA=\dfrac{1}{2}\left(32+48\right)\times 14+64+144}$

= 768 cm^{2}

**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 P_{1 }= 20 cm, P_{2} = 28 cm, l = 9 cm

${\therefore LSA=\dfrac{1}{2}\left( 20+28\right)\times 9}$

= 216 cm^{2}

Last modified on March 28th, 2023

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