Frustum of a Regular Pyramid Formula

A Pyramid is a Mathematical figure having three or four triangular faces as sides and a flat polygonal base which can be triangular, square or rectangular, etc. The side triangular faces are called Lateral faces. The common meeting point of all the triangular faces is called the apex. For a given pyramid having a base with ‘b’ sides has ‘2b’ edges and ‘b + 1’ faces and vertices.

fig. 1: Pyramid

fig. 2: Pyramid

Frustum of a Regular Pyramid Formulae

The Frustum of a Regular Pyramid is formed when the top portion with the apex of the pyramid is cut-off. The remaining geometrical figure formed after chopping off the pyramid is called a frustum. The frustum so formed has 2 bases, one being the actual flat base of the pyramid and the other flat base which gets formed when the top portion of the pyramid is separated out. 

fig. 3: A frustum gets formed when the top portion of the pyramid is chopped off

fig. 4: A frustum gets formed when the top portion of the pyramid is chopped off

They are denoted by ‘b₁‘ and ‘b₂‘. The perpendicular distance between the flat top-base of the frustum and the flat bottom-base of the frustum is called height and is denoted by ‘h’. Likewise, the slant height between the two bases of the frustum is denoted by ‘s’. The lateral surface area and volume of the frustum can be calculated after knowing the values of areas of two bases and height for the volume of the frustum and the values of the perimeter of the two bases and slant height for finding the lateral surface of the surface. We will see how both can be calculated by a simple formula.

Volume of Frustum

Volume of Frustum is given as,

(b₁ + b₂ + (b₁ × b₂)^1/2) × h/3

Here,

b₁ is an area of first base.

b₂ is Ares of second base.

h is the height of the frustum.

Lateral Surface Area of Frustum

Lateral Surface Area of Frustum is given as,

(p₁ + p₂) × s/2

Here,

p₁ is the perimeter of first base

p₂ is the perimeter of the second base

s is the slant height of the frustum

Sample Problems

Question 1: The area of base b₁ of Frustum is 80 m² and the area of base b₂ of Frustum is 20 m². If the height of the frustum is 3 m, what will be the volume of the Frustum.

Solution: 

We know, Volume of Frustum = (b₁ + b₂ + (b₁ × b₂)^1/2) × h/3

Given b₁ = 80 m²

b₂ = 20 m²

h = 30 m

Putting values in the given Volume of Frustum formula,

(b₁ + b₂ + (b₁ × b₂)^1/2) × h/3

= (80 +  20 + (80 × 20)^1/2) × 30/3

= (100 + (1600)^1/2) × 1

= 100 + 40

= 140 m³

Question 2: The perimeter of the base₁ of the frustum is 50 m and the perimeter of the base₂ of the frustum is 10 m. If the slant height of the frustum is 6 m, what will be the lateral surface area of the Frustum.

Solution: 

We know, Lateral Surface Area of Frustum = (p₁ + p₂) × s/2 

Given p₁ = 50 m

p₂ = 10 m

s = 6 m

Putting values in the given Lateral Surface Area of Frustum 

= (p₁ + p₂) × s/2

= (50 + 10) × 6/2

= 60 × 6 / 2

= 180 m

Question 3: The area of base b₁ of Frustum is 25 m² and the area of base b₂ of Frustum is 4 m². If the height of frustum is 0.3 m, what will be the volume of Frustum.

Solution:

We know, Volume of Frustum = (b₁ + b₂ + (b₁ × b₂)^1/2) × h/3

Given b₁ = 25 m²

b₂ = 4 m²

h = 0.3 m

Putting values in the given Volume of Frustum formula

(b₁ + b₂ + (b₁ x b₂)^1/2) × h/3

(25 +  4 + (25 x 4)^1/2) × 0.3/3

= (29 + 10)/ 10

= 39/10

= 3.9 m³

Question 4: The perimeter of the base₁ of the frustum is 45 m and the perimeter of the base₂ of the frustum is 15 m. If the slant height of the frustum is 5 m, what will be the lateral surface area of the Frustum.

Solution: 

We know, Lateral Surface Area of Frustum = (p₁ + p₂) × s/2

Given p₁ = 45 m

p₂ = 15 m

s = 5 m

Putting values in the given Lateral Surface Area of Frustum

= (p₁ + p₂) × s/2

= (45 + 15) × 5/2

= 60 × 5 / 2

= 30 × 5

= 150 m

Question 5: The area of base b₁ of Frustum is 10 m² and the area of base b₂ of Frustum is 40 m². If the height of the frustum is 6 m, what will be the volume of the Frustum.

Solution:

We know, Volume of Frustum = (b₁ + b₂ + (b₁ × b₂)^1/2) × h/3

Given b₁ = 10 m²

b₂ = 40 m²

h = 6 m

Putting values in the given Volume of Frustum formula

(b₁ + b₂ + (b₁ × b₂)^1/2) × h/3

(10 +  40 + (10 × 40)^1/2) × 6/3

= (50 + 20) × 2

= 140 m³

Question 6: The perimeter of the base₁ of the frustum is 75 m and the perimeter of the base₂ of the frustum is 25 m. If the slant height of the frustum is 7 m, what will be the lateral surface area of the Frustum.

Solution: 

We know, Lateral Surface Area of Frustum = (p₁ + p₂) × s/2

Given p₁ = 75 m

p₂ = 25 m

s = 7 m

Putting values in the given Lateral Surface Area of Frustum

= (p₁ + p₂) × s/2

= (75 + 25) × 7/2

= 100 × 7 / 2

= 100 × 3.5

= 350 m

Question 7: The area of base b₁ of Frustum is 80 m² and the area of base b₂ of Frustum is 20 m². If the height of the frustum is 3 m, what will be the volume of the Frustum.

Solution:

We know, Volume of Frustum = (b₁ + b₂ + (b₁ × b₂)^1/2) × h/3

Given b₁ = 80 m²

b₂ = 20 m²

h = 30 m

Putting values in the given Volume of Frustum formula

(b₁ + b₂ + (b₁ x b₂)^1/2) × h/3

= (80 +  20 + (80 x 20)^1/2) × 30/3

= (100 + (1600)^1/2) × 1

= 100 + 40

= 140 m³