Frustum of Cone: Definition, Surface Area & Volume, Formulas

Frustum of cone is the part of a crown that is obtained by cutting it into two plains.. The upper part of the cone remains of the same shape while the lower part makes a frustum.

Hence frustum of cone may be defined as the portion of the cone between the cutting plane and the base of the cone.

Keywords : Frustum, cone, area, pyramid, perimeter, circumference, volume, radius. 

Also read: Quadrilateral Formula

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Frustum of Cone

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When a solid, generally a cone, is cut in such a way that the base of the solid and the plane that cuts the solid are parallel to each other, part of the solid remaining between the parallel cutting plane and the base is known as a frustum of that solid. 

Depending on the shape from which it is obtained, there are three different types of frustum -

1. Frustum of cone
2. Frustum of a triangular pyramid
3. Frustum of a square pyramid 

The frustum of a cone is the part of the cone when it is split into 2 elements with a plane that's parallel to the bottom of the cone. 

Example - An ice-cream cone completely filled with ice cream. When the cone is sliced, the section left between the base and parallel plane is the frustum.

The video below explains this:

Surface Area and Volume Detailed Video Explanation:

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Surface Area of a Frustum of Cone 

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There are two types of areas in frustum of cone - 

• Curved Surface Area (CSA) or Lateral Surface Area (LSA): Area of the curved surface of the frustum.
• Total Surface Area (TSA): Sum of areas of all its faces (i.e., CSA + Sum of areas of the circular bases).

CSA of frustum of cone = πl [ (R² - r²) / r ] (OR) πL (R + r)

Total surface area (TSA) = Area of bases + Lateral (or curved) surface area

CSA of frustum of cone = πl [ (R² - r²) / r ] (OR) πL (R + r)

We know that the base areas of the frustum of the cone are πR² and πr². 

Thus, the sum of base areas is π (R² + r²).

TSA of frustum of cone = π (R² + r²) + πl [ (R² - r²) / r ] 

Or

TSA of frustum of cone = πL (R + r) + π (R² + r²)

There are two methods to derive the curved surface area of a frustum of cone formula.

Method 1 

The circumference of frustum of cone are 

C₁= 2πR

C₂ = 2πr

Substituting these values in the curved surface area of frustum formula,

CSA = (1/2) ×( C₁+C₂)(C₁+C₂) × L

CSA = (1/2) × (2πR + 2πr) × L

CSA = πL (R + r)

Method 2 

The curved surface area of the full cone is πR(L + l).

The curved surface area of the cone (with apex) that is cut is, πrl.

The curved surface area (CSA) of frustum of the cone = Curved surface area of the full cone - Curved surface area of the cone that is cut

CSA = πR (L + l) - πrl... (1)

(L + l) / l = R / r ... (2)

L + l = Rl / r ... (3)

Substituting this in (1),

CSA = πR (Rl / r) - πrl

V = πl [ (R² - r²) / r ]

From (2),

(L / l) + 1 = R / r

L / l = (R / r) - 1

L / l= (R - r) / r

Reciprocating on both sides,

l / L = r / (R - r)

l = (L r) / (R - r)

Substituting this in the above formula,

CSA = (π) [ (L r) / (R - r) ] [ (R² - r²) / r ]

Using the algebraic formulas, a² - b² = (a - b) (a + b) and by applying this formula to R² - r²,

CSA = (π ) [ (L r) / (R - r) ] [ (R - r) (R + r) / r ]

CSA = πL (R + r)

Also read: Surface area of right sided cone 

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Volume of Frustum of Cone 

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There are two methods to get the volume of the cone. 

Method 1

The base areas of the frustum of the cone are:

S1 = πR²

S2 = πr²

Substituting these values in the volume of frustum formula,

V = H3(S1+S2+√S1S2)

H3(S1+S2+S1S2)

V = H/3(πR²+πr²+√πR²⋅πr²)

V=πH/3(R² + Rr + r²)

Method 2

The volume of the full cone is πR² (H + h) / 3.

The volume of the cone (with apex) = πr²h / 3.

We have,

The volume of frustum of the cone( V) = The volume of the full cone - The volume of the cone that is cut

V = πR² (H + h) / 3 - πr²h / 3 ... (1)

The triangles OBC and PQC are similar,and thus,

(H + h) / h = R / r ... (2)

H + h = Rh / r ... (3)

Substituting this in (1),

V = πR² · (Rh / r) - πr²h / 3

V = πh/3 [ (R³ - r³) / r ]

We derived one formula of volume of the frustum of the cone. 

Now we will derive another formula from this.

From (2),

(H / h) + 1 = R / r

H / h = (R / r) - 1

H / h = (R - r) / r

Reciprocating on both sides,

h / H = r / (R - r)

h = (H r) / (R - r)

Substituting this in the above formula,

V = (π / 3) [ (H r) / (R - r) ] [ (R³ - r³) / r ]

Using one of the algebraic formulas, a³ - b³ = (a - b) (a² + ab + b²). 

By applying this formula to R³ - r³,

V = (π / 3) [ (H r) / (R - r) ] [ (R - r) (R² + Rr + r²) / r ]

V = πH/3 (R² + Rr + r²)

Also read: Unit Conversion

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Important formula

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1. Total surface area (TSA) = Area of bases + Lateral (curved) surface area
2. The volume of right circular cone = 1⁄3 r²h
3. Volume of frustum of cone = V=πh/3(R² + Rr + r²)

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Things to remember

• A cone has three parts- The curved portion, apex which is pointed at the top and the base which is the flat portion at the bottom. 
• The curved surface area of frustum of a cone: CSA = πl(R – r)
• The lateral surface of the cone is the same as its curved surface area.
• Examples of a frustum of a cone include- Glass tumbler, space capsule, lampshade, and buckets. 

Also read: Isosceles Triangle Theorems