Solved Examples on Frustum of Cone

Example 1: Find out the Volume of a frustum of a cone that is 15 cm high and the radii for both the bases are 5 cm and 8 cm. 

Solution: 

Using the formula studied above, one can write,

V = 1/3 πH(r² + r’² + rr’)

Given, 

H = 15 cm
r’= 5 cm
r = 8 cm

V = 1/3 π15(8² + 5² + 40)

V = 5π(129)

V = 645π cm³

Example 2: Find out the surface area and total surface area of a frustum of a cone which is 10 cm high and the radii for both the bases are 4 cm and 8 cm. 

Solution: 

We know the formula for surface area and total surface area of the frustum. We need to plug in the required values. 

Curved Surface area of the frustum = πl(r+r’)

where,
L = √ [H² + (R – r)²]

Given,
H = 10 cm
r = 4 cm
R = 8 cm 

Calculating the value of L, 

L = √ [10² + (8 – 4)²]

  =  √(100+16) =  √(116)

Curved Surface Area of Frustum = πL(R+r)

= π√(116)×(8+4)

= 48π√(29)

Total Surface Area = Curved Surface Area of Frustum + Area of Both Bases

= 48π√(29) + π(8)² + π(4)² 

= 48π√(29) + 64π + 16π

= 48π√(29) + 80π cm²

Example 3: Let’s say we have an open metal bucket whose height is 50cm and the radii of the bases are 10cm and 20cm. Find the area of the metallic sheet used to make the bucket.

Solution: 

Bucket is in the form of frustum which closed from the bottom. We need to calculate the total surface area of this frustum. 

Given
H = 50 cm
r ‘= 10 cm
r = 20 cm

Curved Surface Area of Frustum = πL(R+r)

L = √ [H² + (r – r’)²]

L = √ [50² + (20 – 10)²]

  = √(2500+100) =  √(2600)

  = √100(26) = 10√(26)

Curved Surface Area of Frustum = πL(R+r)

= π10√(26)×(20+10)

= 300π√(26)

Total Surface Area = Curved Surface Area of Frustum + Area of Both Bases

= 300π√(26) + π(20)² + π(10)² 

= 300π√(26) + 400π + 100π

= (300π√(26) + 500π) cm²

Example 4: Find out the expression of the volume for a frustum if its height is 6y, and its radii are y and 2y respectively.

Solution: 

Using the formula studied above, 

V = 1/3 πH(r² + r’² + rr’)

Given, 

H = 6y
r’= y
r = 2y

V = 1/3 π6[(2y)² + (y)² + (y)(2y)]

V = 2πy(7y²)

V = 14πy³ unit³

Frustum of a cone is a special shape that is formed when we cut the cone with a plane parallel to its base. The cone is a three-dimensional shape having a circular base and a vertex. So the frustum of a cone is a solid volume that is formed by removing a part of the cone with a plane parallel to circular base. The frustum is not only defined for cones but can be also defined for the different types of pyramids (square pyramid, triangular pyramid, etc.).

Some of the common shapes of a frustum of cone which we discover in our daily life are buckets, lamp shade, and others. Let us learn more about the frustum of cones in this article.