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Volume of Frustum of Cone

Some important terms related to the frustum of a cone. These terms will be used in the frustum of the cone formula.

• Height: The perpendicular distance between the two circular bases is the height of the frustum of a cone. From the above figure, ‘h’ is the height of the frustum of a cone.

• Slant Height: The slant height of a frustum of a right circular cone is the line segment joining the extreme point of two parallel radii, drawn in the same direction, of the two circular bases.

Now let us derive the frustum formula for volume. The frustum of cone formula will be very useful to find the volume of combined figures.

 

To Calculate the Volume of Frustum of Cone

We can calculate the volume of frustum of a cone by subtracting the volume of the smaller cone from the volume of the larger cone.

The larger cone is considered as cone 1 and smaller cone as cone 2.

 

Let us consider h as the height, l as the slant height, and r  as the radius of the larger cone 1.

And consider h’ as the height, l’ as the slant height and r’  as the radius of the smaller cone 2.

 

The height of frustum be H and its slant height is L.(image will be updated soon)

Now, The Volume of Frustum of Cone = Volume of right circular cone 1  – Volume of smaller cone 2

Volume of right circular larger cone 1 = ⅓ π r2 h

Volume of right circular smaller cone 2 = ⅓ π r’2 h’

Therefore, The Volume of Frustum of Cone V = ⅓ π r2 h – ⅓ r’2 h’

= ⅓  π( r2h – r’2h’)………equation (1)

Now consider  

ΔOO’D and ΔOPB

∠DOO’ = ∠BOP…( common angles)

As CD ll AB

∠O’DO= ∠PBO ….(Corresponding Angles)

Therefore, ΔOO’D ΔOPB ( by AA criteria)

Hence, according to similar triangles property,  the ratio of corresponding sides must be equal.

We get h’/h = r’/r

h = h’r/r’……..equation (2)

 

Substitute the value of equation (2) in equation (1) we get 

The Volume of Frustum of Cone V = ⅓ π( r2h – r’2h’2)

  = ⅓ π (r2 (rh’/r’) – r’2h’2)

= ⅓ π ( r3h’/ r’ – r’3h’2/ r’)

= ⅓ π h’ ( r3 – r’3/ r’)

 

From the figure we get,

h = H + h’…………………equation (3)

Substitute in equation (2)

h’/h = r’/r

h’/H + h = r’/r

h’ = H(r’/r -r’)

Substitute this in 

The Volume of Frustum of Cone V = ⅓ π h’ ( r3 – r’3/ r’)

= ⅓ π[ H ( r’/ r-r’)] ( r3 – r’3/ r’)

=⅓ π H (r2 + r’r + r’2)

This is the frustum formula for volume

 

Similarly, we can calculate the surface area of the frustum of a cone.

Curved surface area of larger right circular cone 1 = πrl

Curved surface area of smaller right circular cone 2 = πr’l’

′Curved surface area of the frustum of cone 

= Curved surface area of larger right circular cone 1 – Curved surface area of smaller right circular cone 2 

= πrl − πr’l’

= π ( rl – r’l’)

From fig.we know  ΔOOD ~ ΔOPB

 Therefore l’/l = r’/r……equation (4)

But l’ = l  – L

Substitute in equation (4)

l = L( r / r- r’)

The circumference of the base is the length of arc S and S’

 which is given by

S ′= 2πr

S’ = 2πr’

Curved surface area of frustum = A = ½ ( Sl- S’l’)

A = ½ x 2πrl − ½ x 2πr’(lr’r)

= πl( r- r’2/r)

= πl( r2– r’2/r)

Substituting the value of l

l = L( r / r- r’)

Curved surface area of frustum A = π L ( r/ r-r’)(r2 – r’²/r)

This is the frustum formula for curved surface area

 

The total surface area is given as the sum of the curved surface area and the area of the base. Hence the total surface area of the frustum is:

 

Solved Examples

Example 1: If the radii of the circular ends of a conical bucket which is 45 cm high are 28 cm and 7 cm, find the capacity of the bucket (Use π= 22/7).

Solution: Clearly, bucket forms a frustum of a cone such that the radii of its circular ends are r1 = 28 cm, r2 = 7 cm, and height h = 45 cm. 

Let V be the capacity of the bucket.

 Then, by frustum of cone formula

 

V = Volume of the frustum

V = ⅓ πh(r12 + r22 + r1r2)

V = ⅓ x 22/7 x 45 (28 x 4 + 7 + 28) 

   = 330 x 147 cm3 

   = 48510 cm3.

 

Example 2: The slant height of the frustum of a cone is 4cm, and the perimeter of its circular bases is 18cm and 6cm respectively. Find the curved surface area of the frustum.

Solution: let r1 and r2  be the radii of the circular base of the frustum, l be the slant height, and h be the height.

We have l = 4cm, 

2 π r1 = 18, r1 = 9/π

and 

2 π r2 = 6 , r2 = 3/π 

By frustum of a cone formula

Curved surface area = π( r1+ r 2)l 

=  π(  9/𝜋 + 3/𝜋 ) x 4

=  48 cm2

Quiz Time

1. The slant height of the frustum of a cone is 8cm and the perimeters of its circular ends are 20cm and 24cm respectively. Find the curved surface area of a cone.

2. The radii of the circular ends of a frustum of height 6cm are 14cm and 6cm respectively. Find the lateral surface area and surface area of the frustum.