104 LearnersLast updated on June 24th, 2025
Volume of Cone
A cone is a three-dimensional shape with a circular base and a single pointed tip called the vertex. Some common examples include traffic cones and party hats. A cone is formed by rotating a right-angled triangle around one of its sides. The volume of a cone refers to the amount of space it takes up. In this topic, we will explore how to calculate the volume of a cone.
What is the Volume of a Cone?
The volume of a cone is measured in cubic units such as cm3, m3, in3, etc
The cone’s structure includes both a curved surface area and a flat circular base. The base is connected to the vertex by every point on it’s surface area.
The volume of a cone is calculated using its radius and height.
Volume of Cone Formula
The volume of a cone is the product of one-third of the area of its base and its vertical height. Mathematically, the formula is written as V = 13r2h cubic units.
Here, r is the radius of the base, h is the perpendicular height from the base to the vertex, and the value of =3.14 or 22/7.
If the diameter of the cone is given but not the radius, the radius of the cone can be found by dividing the diameter by 2.
The volume of cone can also be found using formula V=1/12 πd2h.
This formula is found by substituting the valure of r with d/2:
V = 1/3πr2h
Substituting r = d/2
V = 1/3π(d/2)2h
V = 1/3(d2/4)h
V =1/12 π d2h
In case the height of the cone is not given, but the slant height is, we can find the height using Pythagorean theorem i.e., h=√l2-r2
Here, h is the height of the cone
l is the slant height, and
r is the radius
How to Derive the Volume of Cone Formula?
To derive the formula for the volume of a cone, we start by considering a cylinder with the same height (h) and radius (r) as the cone. The height and radius are essential because the volume of a cone depends on the area of its circular base (which uses the radius) and how tall the cone is (its height), both of which directly affect the space it occupies
When we try to fill the cylinder using the cone, we find that a total of 3 cones are required to fill one cylinder.
Since we already know the volume of a cylinder = πr2h, and we have established that the volume of a cone having the same radius is one-third the volume of a cylinder.
So, Volume of cone =1/3πr2h.
How to Find the Volume of Cone?
The volume of a cone can be found by substituting the values of the required parameters given in the formula.
- Make a note of all known parameters: the radius ‘r’, diameter ‘d’, slant height ‘l’ and height ‘h’.
- Apply one of the two formulas depending on the given parameters:
Volume of cone using radius: V=1/3πr2h or V=1/3π r2 √l2-r2
Volume of cone using diameter: V=1/12πd2h
- Write the final answer in a cubic unit.
Tips and Tricks for Calculating the Volume of Cone
Understanding the volume of a cone becomes easier with the help of some useful tips. These tricks can help students avoid confusion, remember key steps, and solve problems more efficiently.
- Always make sure that units are uniform throughout, i.e., check that the radius and height are of the same unit.
- Volume should always be in cubic units.
- Use =22/7 when the radius is divisible by 7, and use =3.14 for other values of the radius. This makes calculations easier.
- Draw and label the diagram of a cone for visual help while solving for its volume.
- Follow the process step by step to avoid missing steps like squaring.
Common Mistakes and How to Avoid Them in Volume of Cone Calculations
When calculating the volume of a cone, along with the formula, it is necessary to pay attention the given values and unit conversion. Here are a few common mistakes to avoid:
Mistake 1
Using diameter instead of radius
Students may directly use the diameter given in the question without solving for the radius first. Always divide the diameter by 2 to get the radius and solve for the volume.
Mistake 2
Using the wrong formula
The volume of a cone is = 1/3πr2h; some may confuse it with the volume of a cylinder, which is πr2h.
Mistake 3
Using the value of slant height instead of vertical height
If only the slant height is given, use the formula h= √l2-r2 to find the vertical height before substituting it in the formula.
Mistake 4
Incorrect units or unit conversion
Students may forget to convert units, for example, from cm to m, give the answer without specifying units, or write the wrong unit. So, always check for unit consistency.
Mistake 5
Forgetting to square the radius
While applying the formula V = 1/3 πr2h, always square the radius to get the right answer.
Volume of Cone Examples
Problem 1
A cone has a radius of 3 cm and a height of 4 cm. Find its volume.
37.68 cm3
Explanation
Volume of a cone = 1/3 πr2h
=1/3 x 3.14 x 32 x4
=1/ 3 x 3.14x 9 x 4
=1/3 x 113.04
=37.68 cm3
Problem 2
A cone has a diameter of 10cm and a height of 12 cm. Find its volume
314cm3
Explanation
Radius r=10/2 = 5cm
Volume = 1/3 πr2h
=1/3 x 3.14 x 52 x 12
=1/ 3 x 3.14 x 25 x 12
= 1/3 x 942
=314 cm3
Problem 3
A cone has a radius of 2.5 cm and a height of 6 cm. Find the volume.
39.25 cm3
Explanation
Volume = 1/3 πr2h
=1/3 x 3.14 x (2.5)2 x 6
= 1/3 x 3.14 x 6.25 x 6
=1/3 x 117.75
=39.25 cm3
Problem 4
Find the volume of a cone with radius 7 cm and height 15 cm. Use =22/7
770 cm3
Explanation
Volume = 1/3 πr2h
=1/3 x 22/7 x 72 x 15
= 1/3 x 22/7 x 49 x 15
=1/ 3 x 16170/7
= 16170/21
= 770 cm3
Problem 5
A birthday party hat is shaped like a cone with a base radius of 6 cm and a height of 10 cm. What is its volume?
376.8 cm3
Explanation
Volume = 1/3 πr2h
= 1/3 x 3.14 x 62 x 10
=1/3 x 3.14 x 36 x 10
= 1/3 x 1130.4
=376.8 cm3
FAQs on Volume of Cone
1. What is a cone?
2.What is the formula for the volume of a cone?
3.What units are used for the volume of a cone?
4. How do you find the height of a cone if you know its radius and volume?
5. What is the difference between the height and the slant height of a cone?
6.How can children in Vietnam use numbers in everyday life to understand Volume of Cone ?
7.What are some fun ways kids in Vietnam can practice Volume of Cone with numbers?
8.What role do numbers and Volume of Cone play in helping children in Vietnam develop problem-solving skills?
9.How can families in Vietnam create number-rich environments to improve Volume of Cone skills?
Important Glossaries for Volume of Cone
- Vertex: A vertex is the tip or apex of the cone that is opposite to its base.
- Curved Surface Area: It is the area covered by the curved surface of a cone, it does not include the base of a cone since the base is a flat surface.
- Radius: It is the distance from the center of the base of a cone to its edge.
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
