# Truncated Cone (Frustum of a Cone)

A truncated cone is a cone that is chopped off from the tip with a plane parallel to its base.

The diagram below shows how a full cone is truncated to form a frustum.

## Parts

1. 2 Bases – The circular ends (flat faces), one at the top, and one at the bottom.
2. 1 Curved side face – The lateral face bounding the circular bases.
3. 2 Radii – The radius of the larger circular base is the big radius (R), and the radius of the smaller circular base is the little radius (r).
4. Height – The distance between the centers of the 2 bases. It is represented by ‘h’.
5. Slant height – The shortest distance between the outer edges of the bases. It is represented by ‘l’.

## Formulas

### Volume

The formula is:

Find the volume of a truncated cone with radii of 10 cm and 7 cm, and a height of 13 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}\pi h\left( R^{2}+r^{2}+R\times r\right)}$, here R = 10 cm, r = 7 cm, h = 13 cm, π = 3.141 = ${\dfrac{22}{7}}$
∴ V = ${\dfrac{1}{3}\times 3.141\times 13\times \left( 10^{2}+7^{2}+10\times 7\right)}$
= 2,981.37 cm3

### Surface Area

The formula is:

Find the surface area of a truncated cone with a base radius of 7 cm, a top radius of 3 cm, and a slant height of 12 cm.

Solution:

As we know,
Surface Area (SA) = ${\pi \left( R+r\right) l+\pi \left( R^{2}+r^{2}\right)}$, here R = 7 cm, r = 3 cm, l = 12 cm, π = 3.141 = ${\dfrac{22}{7}}$
∴ SA  = ${3.141\times \left( 7+3\right)\times 12+3.141\times \left( 7^{2}+3^{2}\right)}$
= 559.2 cm2

We also can find the lateral or curved surface area. The formula is:

Lateral Surface Area (LSA) = ${\pi \left( R+r\right) l}$, here R = base radius, r = top radius, l = slant height

Let us solve an example to understand the concept better.

Find the lateral surface area of a truncated cone using the formula given in the figure alongside. The values are: R = 11 cm, r = 5 cm, and a l = 8 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\pi \left( R+r\right) l}$, here R = 11 cm, r = 5 cm, l = 8.5 cm, π = 3.141 = ${\dfrac{22}{7}}$
∴ LSA  = ${3.141\times \left( 11+5\right)\times 8.5}$
= 427.2 cm2

Finding the lateral and total surface area when the BASE, RADII, and HEIGHT are known

Calculate the lateral and total surface area of a truncated cone with radii 6 mm, 2 mm, and a height of 9 mm.

Solution:

Here, we will apply an alternative formula.
Lateral Surface Area (LSA) = ${\pi \left( R+r\right) \sqrt{\left( R-r\right) ^{2}+h^{2}}}$, here R = 6 mm, r = 2 mm, h = 9 mm, π = 3.141
∴ LSA  = ${3.141\times \left( 6+2\right) \sqrt{\left( 6-2\right) ^{2}+9^{2}}}$
= 247.53 mm2
Total Surface Area (TSA) ${=\pi \left( R^{2}+r^{2}\right) +LSA}$, here R = 6 mm, r = 2 mm, LSA = 247.53 mm2
${\therefore TSA=3.141\times \left( 6^{2}+2^{2}\right) +247.53}$,
= 373.17 mm2