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Volume of Cone | Formula, Derivation and Examples

Last Updated : 29 Feb, 2024
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Volume of cone can be defined as the space occupied by the cone. As we know, cone is a three-dimensional geometric shape that has a circular base and a single apex (vertex).

Volume of Cone

Let’s learn about Volume of Cone in detail, including its Formula, Examples and Frustum of Cone.

What is Volume of Cone?

A cone’s volume is defined as the amount of space or capacity it fills. The volume of a cone is measured in cubic units such as cm3, m3, in3, and so on. By rotating a triangle around any of its vertices, a cone can be produced. Volume of a cone can also be measured in litres.

  • A cone may be divided into two types: right circular cones and oblique cones.
  • The vertex of the right circular cone is vertically above the centre of the base, but the vertex of the oblique cone is not vertically above the centre of the base.
  Formulas Related to Volume of Cone
Volume of a Cone V = 1/3 πr2h = = (1/12)πd2h
Volume of a Cone (Slant Height) V = 1/3 πr2(√{L2 – r2})
Volume of Frustum of a Cone 1/3 π h [{r3 – (r’)3} / r]
Volume of a Cone (Doubled Radius and Height) V = (8/3)πr2h
Volume of a Cone (Halved Radius and Height) V = (1/24)πr2h

Volume of Cone Formula

A cone is a solid three-dimensional form having a circular base. It has a curved surface. The perpendicular height is the distance from the base to the vertex.

Formula of Volume of Cone :

V = 1/3 πr2h

where,

  • r is Radius of the Cone
  • h is Radius of the Cone
  • π is constant with value 22/7 or 3.14

Slant Height of Cone

Slant height of cone is the distance from its apex (top point) to any point on the perimeter of its circular base. It is the straight-line distance along the lateral surface, not through the interior of the cone.

Slant height of a cone can be derived using the Pythagoras theorem ,

h2 + r2 = L2

h = √(L2 – r2)

Cone Volume in terms of Slant Height

For a cone with height ‘h’ and radius ‘r’ the slant height ‘L’ of the cone is given by the formula,

h2 + r2 = L2

h = √(L2 – r2)…(i)

Then the volume of the cone in terms of slant height is,

V = (1/3)πr2h…(ii)

Using the value of h in the eq (ii), we get the formula for volume of cone as,

V = (1/3)πr2√(L2 – r2)

Volume of Cone Derivation

Let’s suppose we have a cone with a circular base whose radius is r and height is h.

Volume of Cone Derivation

We know that the volume of a cone is equal to one-third of the volume of a cylinder having the same base radius and height.

So, the volume becomes,

V = 1/3 × Circular Base Area × Height

V = 1/3 × πr2 × h

V =  πr2h/3

This derives the formula for the volume of a cone.

How to Find the Volume of Cone?

Let’s consider an example to determine the volume of a cone.

Example: Determine the volume of a cone if the radius of its circular base is 3 cm and the height is 5 cm.

Step 1: Note the radius of the circular base (r) and the height of the cone (h). 

Here, the radius is 3 cm and the height is 5 cm.

Step 2: Calculate the area of the circular base = πr2. Substitute the value of r and π in the given equation, 

i.e., 3.14 × (3)2 = 28.26 cm2.

Step 3: We know that the volume of a cone is (1/3) × (area of the circular base) × height of the cone. 

Then, substitute the values in the equation = (1/3) × 28.26 × 5 = 47.1 cm3.

Step 4: Hence, the volume of the given cone is 47.1 cm3.

Using the steps discussed above the volume of a cone can be calculated.

Volume of Cone with Height and Radius

The volume of the cone if its height(h) and radius(r) are given is calculated using the formula,

V = (1/3)πr2h cubic units

Volume of Cone with Height and Diameter

Volume of Cone when the diameter and height of cone is given is calculated below. Let us suppose we are given a cone with radius “r” and diameter “d”.

Then the radius of the base is half of the diamter of the base, i.e. r = d/2

Volume of the cone if its height(h) and diameter(d) are given is calculated using the formula,

V = (1/12)πd2h cubic units

Volume of Cone (If Radius and Height are doubled)

Suppose,

  • Radius of the Cone (r) = 2r
  • Height of the Cone (h) = 2h

Then the volume of a cone is given as,

Volume of a Cone = (1/3)π(2r)2(2h) cubic units

V = (⅓)π(4r2)(2h)

V = (8/3)πr2h

Thus, volume of a cone becomes 8 times the original volume i.e. V = (8/3)πr2h, when its radius and height are doubled.

Cone Volume (If Radius and Height are Halved)

Let us suppose,

  • Radius of the cone (r) = r/2
  • Height of the Cone (h) = h/2

Then the volume of a cone is given as,

Volume of a Cone = (1/3)π(r/2)2(h/2) cubic units

V = (1/3)π(r2/4)(h/2)

V = (1/24)πr2h

Thus, volume of a cone becomes 1/8 times the original volume i.e. V = (1/24)πr2h, when its radius and height are halved.

Frustum of Cone

Frustum is the sliced part of a cone, and the volume of the frustum of the cone is the amount of liquid any frustum can hold.

So for calculating the volume, we need to find the difference in the volumes of two cones. 

Volume of Frustum of Cone

Formula of volume of the frustum of cone is given by subtracting the volume of the smaller cone from the bigger one.

Frustum of Cone Volume

From the above figure, we have, 

  • Total height H’ = H + h
  • Slant height L = l1 + l2
  • Radius of Cone = r
  • Radius of the sliced cone = r’

Now the volume of the bigger cone = 1/3 π r2 H’ = 1/3 π r2 (H+h)

Volume of the smaller cone = 1/3 π(r’)2h. The volume of the frustum can be calculated by the difference between the two cones, i.e.

Volume of Frustum = 1/3 π r2 H’ -1/3 π(r’)2h

V = 1/3π r2 (H+h) – 1/3 π(r’)2h

v = 1/3 π [ r2 (H+h) – (r’)2 h ] ………(1)

Using the properties of similar triangles in Δ QPS and Δ QAB. we have, 

r/ r’ = H+h / h

H+h = (rh)/r’

Substituting the value of H+h in the formula for the volume of frustum we get,

Volume of Frustum = 1/3 π [r2 (rh/r’) – (r’)2 h] 

V = 1/3 π  [r3h/r’ – (r’)2 h]

V = 1/3 π h (r3/r – (r’)2)  

V = 1/3 π h [{r3 – (r’)3} / r]

Volume of Frustum of Cone = 1/3 π h [{r3 – (r’)3} / r]

where,

  • r is Radius of the Lower Base of Frustum of Cone
  • r’ is Radius of the Upper Base of Frustum of Cone
  • h is Height of the Smaller Cone
  • π is constant with value 22/7 or 3.14

Related :

Volume of Cone Examples

Let solve some questions on the Volume of Cone formulas.

Example 1. Find the volume of a cone for a radius of 7 cm and height of 14 cm.

Solution:

We have,

  • r = 7
  • h = 14

Volume of Cone = 1/3 πr2h

V = (1/3) (22/7) (7) (7) (14)

V = (1/3) (7) (7) (2)

V = 32.66 cm3

Example 2. Find the volume of a cone for a radius of 5 cm and height of 9 cm.

Solution:

We have,

  • r = 5
  • h = 9

Volume of Cone = 1/3 πr2h

V = (1/3) (3.14) (5) (5) (9)

V = (3.14) (5) (5) (3)

V = 235.49 cm3

Example 3. Find the volume of a cone for a radius of 7 cm and height of 12 cm.

Solution:

We have,

  • r = 7
  • h = 12

Volume of Cone = 1/3 πr2h

V = (1/3) (22/7) (7) (7) (12)

V = (22) (7) (4)

V = 616 cm3

Example 4. Find the volume of a cone for a radius of 8 cm and height of 15 cm.

Solution:

We have,

  • r = 8
  • h = 15

Volume of Cone = 1/3 πr2h

V = (1/3) (22/7) (8) (8) (15)

V = (1/3) (22/7) (8) (8) (5)

V = 335.02 cm3

Practice Questions on Volume of Cone

Q1. Find the radius of a cone if its volume is 121 cm2 and its height is 2 cm.

Q2. Find the volume of a cone for the height of 12 cm and the slant height of 7 cm.

Q3. Find the volume of a cone for the height of 21 cm and the diameter of base is 12 cm.

Q4. Find the volume of a cone for a radius of 12 cm and height of 5 cm.

Cone Volume -FAQs

Define Volume of Cone.

Volume of a cone is defined as the total capacity of the liquid a cone can hold in 3-dimension. It is the total space occupied by the cone.

What is Volume of Cone Formula?

Volume of a cone is given by the following formula,

Volume of Cone = ⅓ πr2 h cubic units.

How to Find Cone Volume with Slant Height?

The volume of the cone if its slant height(L) and its radius(r) is given is calculated using the formula, V = (1/3)πr2√(L2 – r2)

What is the Total Surface Area (TSA) of Cone Formula?

Total surface area of a cone is given by the formula, TSA of Cone = πr(l + r) square units.

What is Relation between Volume of Cylinder and Cone?

Volume of Cone is 1/3 the Volume of Cylinder.

What is Slant Height of Cone Formula?

The slant height(l) of a cone is calculated using the formula, l = √(h2 + r2).

What is Cone Volume, if Height and Diameter are given?

Volume of Cone if its height (h) and diameter of base (d) is given is, V = (1/12)πd2h cubic units.

How To Find the Volume of Liquid in Cone?

Volume of Liquid inside the cone is calculated using the volume of cone foprmula added above.



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