Volume of Cone – Formula, Derivation, Steps & Examples

What is Volume of a Cone?

The volume of a cone is the measure of how much space is enclosed within the cone. A cone is a three-dimensional geometric shape that has a circular base and tapers smoothly to a point called the vertex or apex. The v of a cone represents the amount of space it occupies or the quantity of material it can hold. The volume of a cone is expressed in cubic units.

Read More: How to Find the Area of Polygon?

Volume of Cone Formula

The volume of a cone formula is related to the area of its circular base and its vertical height. The formula for the volume of a cone is represented as

Volume of cone = (1/3)πr²h

Where,

• r = radius of the base of the cone

• h = height of the cone

• (pi) = 3.14159 (approximately)

Enroll Now for Maths Online Classes

Volume of Cone Formula Using Slant Height

In case the height of the cone is not known, but the radius of the base and slant height are given, we can find the height of the cone and substitute its value in the volume formula of the cone as mentioned earlier.

Let’s consider the radius of the base of the cone is r and the slant height is l. We can find the height of the cone using Pythagoras’s theorem. Since the height is the perpendicular distance between the base and vertex, we can calculate the height (h) of the cone using the following formula:

(h)² + (r)² = (l)² 

Or (h)² = (l)² - (r)² 

Or, h = √(l)² - (l) 2 

Now, by substituting this value of ‘h’ into the volume of cone formula, we get:

Volume of cone (V) = (1/3) πr²h

Or V = (1/3) πr² √(l² - r²).

Read More: Volume of Sphere

Derivation of Volume of Cone Formula

The volume of the cone formula is derived using the concept of the cylinder.

Let’s consider a cylinder and a cone, both having the same base radius r and the same height h.

Now, if you fill the cone completely with water and pour it into the cylinder, you will notice that you will need three cones of water to fill the cylinder completely.

This observation shows that:

Volume of cylinder = 3 x Volume of cone

We know the volume of a cylinder is given by V = πr²h.

So, we can write:

πr²h = 3 x volume of cone

or, volume of cone = 1/3 πr²h

How to Find the Volume of Cone

Calculating the volume of a cone using the volume formula becomes easier for your child if they follow the step-by-step process as mentioned below:

• Step 1: Identify the radius.

Note the radius (r) of the cone’s circular base.

• Step 2: Identify the height.

Note the height (h) of the cone. The height is the perpendicular distance from the vertex to the center of the base.

• Step 3: Apply the formula.

Substitute the values into the volume of cone formula:

V = 1/3 πr²h

• Step 4: Calculate the result.

Simplify the expression to get the final value in cubic units.

Read More: Tetrahedron Shape

Volume of Cone Examples

After understanding what the volume of a cone is and the formula for the volume of the cone, students must learn through regular practice how to apply these formulas in solving sums. Here are some solved examples using the volume of cone formula that can help your child get conceptual clarity on this topic.

Volume of Cone Example 1:

Question: Find the volume of a cone whose radius is 4 cm and height is 9 cm.

Solution:

Here

r = 3

h = 9

Using the formula, we get:

V = 1/3 πr²h

= 1/3 π (4)² x 9

= π x 16 x 3

= 48π

= 48 x 3.14

= 150.72

Answer: The volume of the cone is 150.72 cm³.

Volume of Cone Example 2:

Question: Find the volume of a cone with a radius of 3 cm and a slant height of 5 cm.

Solution: Using the formula V = (1/3) πr² √(l² - r²)

We get:

V = 1/3 x π (3)² √((5)² – (3)²)

= 1/3 x π x 9 √(25 - 9)

= π x 3 √(16)

= π x 3 x 4

= 12π

= 12 x 3.14

= 37.68

Answer: The volume of the cone is 37.68 cm³.

Volume of Cone Example 3:

Question: A conical tank has a height of 12 m and a base radius of 5 m. Find how much water it can hold (in liters).

Solution:

V = 1/3 πr²h

= 1/3 π (5)² x 12

= (1/3) x π x 25 x 12

= 100 π

= 100 x 3.14

= 314 cm³.

We know that 1 cm³ = 1000 liters.

So, V = 314 x 1000 = 314000 liters.

Answer: The tank can hold 314000 liters of water.

Volume of Cone Example 4:

Question: Find the volume of a cone with a base diameter of 8 cm and a height of 15 cm.

Solution: Radius of cone (r) = ½ x diameter = ½ x 8 = 4 cm

h = 15 cm

Therefore,

V = 1/3 πr²h

= 1/3 π (4)² x 15

= (1/3) x π x 16 x 15

= 80 π

= 80 x 3.14

= 251.2 cm³.

Answer: The volume of the cone is 251.2 cm³.

Also read: What is a Polyhedron

Strengthen Your Childs Build Confidence in Maths with CuriousJr

Does your child need extra support with maths outside school? Many students struggle with numbers because they lack regular practice or proper guidance. With CuriousJr’s online maths classes, learning becomes easy, engaging, and stress-free.

Here’s how CuriousJr helps your child succeed:

• Step-by-step learning from two dedicated mentors

• Regular practice to boost speed and accuracy

• Interactive lessons that make maths fun

• Progress reports and parent-teacher updates

Each session helps children understand concepts clearly and gain confidence in solving problems independently. Consistent learning improves both grades and interest in the subject. Book a demo class today and give your child the right start to master maths with confidence.