Volume of Cone- Formula, Derivation and Examples

A cone is a three-dimensional solid that has a circular base and a single vertex, with a curved surface joining them, and the perpendicular distance from the base to the vertex is called its height. The volume of a cone is the space enclosed within it and is measured in cubic units such as cm³, m³, or in³.

Cones are of two types:

• Right Circular Cone: The vertex lies directly above the centre of the base.
• Oblique Cone: The vertex is not directly above the centre of the base.

Formula

The volume of a cone is equal to one-third of the product of the area of its circular base and its height. A cone can be considered similar to a pyramid with a circular base. By knowing the radius and height, the volume can be easily calculated.

Formula of Volume of Cone:

V = 1/3 πr²h
Where,

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

Volume of Cone Using Different Parameters

• With Height and Radius
The volume of a cone is calculated using its radius and height:

Volume = (1/3)πr²h cubic units

• With Height and Diameter
Since the diameter (d) is twice the radius (r), the formula becomes:

Volume = (1/12)πd²h cubic units

• With Slant Height
Using the Pythagorean theorem, the relationship between height (h), radius (r), and slant height (L) is:

h² + r² = L²
⇒ h = √(L² − r²)

Substituting the value of h in the volume formula:
Volume = (1/3)πr²√(L² − r²)

Derivation

Take three identical cones. When each cone is filled with water and poured into the cylinder:

• 1 cone fills 1/3 of the cylinder.
• 2 cones fill 2/3 of the cylinder.
• 3 cones fill the cylinder.

Thus, three cones equal one cylinder.

Steps to find volume

The volume of a cone can be calculated using its radius or diameter along with height or slant height.

Step 1: Identify the given values:

• Radius (r) or Diameter (d)
• Height (h) or Slant height (L)

Step 2: Choose the appropriate formula:

• Using radius:
  V = (1/3)πr²h
  or
  V = (1/3)πr²√(L² − r²)

• Using diameter:
  V = (1/12)πd²h
  or
  V = (1/12)πd²√(L² − r²)

Step 3: Substitute the values and calculate.

Step 4: Write the final answer in cubic units (e.g., cm³, m³).

Related Articles

• Frustum of Cone
• Volume of a Cube
• Volume of Cuboid
• Volume of Sphere
• Volume of Cylinder

Solved Examples

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

Solution:

We have,

• r = 7
• h = 14

Volume of Cone = 1/3 πr²h

= (1/3) × 22 × 7 × 14

= (1/3) × 22 × 98

= (1/3) × 2156

= 718.67 cm³

Example 2. Find the Volume of a cone for a radius of 5 cm and a 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 = 235.5 cm³

Example 3. Find the volume of a cone with a radius of 7 cm and a 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 a 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 = 1005.71 cm³

Practice Questions

Q1. Find the radius of a cone if its volume is 121 cm³ and its height is 2 cm.

Q2. Calculate the Volume of a cone for a height of 5 cm and a slant height of 13 cm.

Q3. What will be the Volume of a cone with a height of 21 cm and a base diameter of 12 cm?

Q4. Find the Volume of a cone for a radius of 9 cm and a height of 4 cm.

Answer:-

1. 7.6 cm
2. 752.8 cm³
3. 791.76 cm³
4. 339.12 cm³