Volume of a Right Circular Cone – Definition, Formula, Examples

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What Is the Volume of a Right Circular Cone?

The volume of a right circular cone is the total space occupied by the right circular cone. It is equal to one-third the product of the base area and height.

The formula to find the volume of a right circular cone is $V = \frac{1}{3} \pi r^{2}h$, where r is the radius of the base circle and h is the height of the cone.

A right circular cone is a type of cone in which the axis of the cone is the line joining the vertex (apex) and the midpoint of the circular base. A right circular cone is generated by a revolving right triangle about one of its legs.

Right circular cone

We know that the volume of a cylinder is given by $\pi r^{2}h$, where r is the radius and h is the height. Thus, we can say that the volume of a cylinder is three times that of a cone having the same radius and height. Three such cones can fill up the cylinder.

Volume of a cone and volume of a cylinder comparison

Volume of Right Circular Cone Formula

Volume of a right circular cone $= \frac{1}{3} \pi r^{2}h$

where 

r = radius of the circular base

h = height of the cone


How to Find the Volume of a Right Circular Cone

Let’s understand how to calculate the volume of a right circular cone based on the provided information.

Finding Volume Using Radius and Slant Height of a Cone

If we know the radius and slant height of the cone, we will first find the height using the
formula: $h = \sqrt{l^{2}\;-\;r^{2}}$

Once we know the height, we can calculate the volume as

Volume of the cone $= \frac{1}{3} r^{2} h = \frac{1}{3} \pi r^{2}\sqrt{l^{2}\;-\;r^{2}}$.

This is the formula for the volume of a right circular cone with slant height.

Finding Volume Using Height and Slant Height of a Cone

If we know the height and slant height of the cone, we will first find the radius using the formula:

$r = \sqrt{l^{2}\;-\;h^{2}}$

Once we know the radius, we can calculate the volume as

Volume of the cone $= \frac{1}{3} \pi r^{2}h$ 

Facts about Volume of a Right Circular Cone

  • The volume of a right circular cone is exactly one-third of the volume of a cylinder with the same base radius and height.
  • If we keep the height constant, a cone with a larger base area will have a larger volume, while a cone with a smaller base area will have a smaller volume.

Conclusion

In this article, we learned how to find the volume of a right circular cone, its formula, and different ways of calculating it. Let’s use the formula to solve a few problems and objective questions based on the concept of volume of a right circular cone.

Solved Examples on Volume of a Right Circular Cone

1. A cylinder and a cone have the same radius and same height. If the volume of a cylinder is 1037 cubic inches, what is the volume of the cone?

Solution: 

Volume of cone $= \frac{1}{3}\times$ Volume of cylinder

Volume of cone $= \frac{1}{3} \times 1038$ 

Volume of cone = 346 cubic inches.

2. Find the volume of the right circular cone if the radius is 7 units and height is 9 units. 

Solution: 

Radius of the cone (r) = 7 units

Height of the cone (h) = 9 units

Volume $= \frac{1}{3} \pi r^{2} h$

Volume $= \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times 9$ 

Volume = 462 cubic units.

3. What is the volume of a conical tent whose radius is 14 feet and height is 12 feet?

Solution: 

Radius of the conical tent (r) = 14 feet

Height of the conical tent (h) = 12 feet

Volume of the cone $= \frac{1}{3} \pi r^{2}h$

$V = \frac{1}{3} \times \frac{22}{7} \times 14 \times 14 \times 12$ 

$V = 2464$ cubic feet.

4. If the volume of the right circular cone is 24.5π cubic inches and height is 6 inches, then what will be the radius of the cone?

Solution: 

Volume of the cone = 24.5π cubic inches

Height of the cone (h) = 6 inches

$\frac{1}{3} \pi r^{2} h = 24.5$

$\frac{1}{3} r^{2} \times 6 = 24.5$

$2r^{2} = 24.5$

$r^{2} = \frac{24.5}{2} = 12.25$

r = 3.5 inches

5. The volume of the right circular cone is 980 cubic inches. What is the height of the cone if the base radius is 7 inches?

Solution: 

Volume of the cone = 980 cubic inches

Radius of the cone (r)=7 inches

$\frac{1}{3} \pi r^{2} h = 980$

$\frac{1}{3} \times 7 \times 7 \times h = 980$

$\frac{49h}{3} = 980$

$h = \frac{980 \times 3}{49} = 60$ inches

Practice Problems on Volume of a Right Circular Cone

Volume of a Right Circular Cone – Definition, Formula, Examples

Attend this quiz & Test your knowledge.

1

The slant height of the right circular cone is 13 inches and radius is 5 inches. What is the height of the cone?

12 inches
18 inches
20 inches
24 inches
CorrectIncorrect
Correct answer is: 12 inches
Radius (r) = 5 inches
Slant height (l) = 13 inches
$h = \sqrt{l^{2}\;-\;r^{2}}$
$h = \sqrt{13^{2}\;-\;5^{2}} = \sqrt{169\;-\;25} = \sqrt{144} = 12$ inches
2

Find the volume of a right circular cone in terms of radius if radius is equal to the height.

$\frac{1}{3} \pi r^{2}$
$\pi r^{3}$
$\frac{1}{3} \pi r^{4}$
$\frac{1}{3} \pi r^{3}$
CorrectIncorrect
Correct answer is: $\frac{1}{3} \pi r^{3}$
Let radius of the cone be r units and height of the cone be h units.
Given: r = h
Volume of the cone $= \frac{1}{3} \pi r^{2} r = \frac{1}{3} \pi r^{3}$
3

If the height of the cone is 18 inches and base diameter is 14 inches, then what will be the volume of the cone?

1024 cubic inches
984 cubic inches
964 cubic inches
924 cubic inches
CorrectIncorrect
Correct answer is: 924 cubic inches
Base diameter of the cone (d)=14 inches
Base radius of the cone (r)=7 inches
Height of the cone (h)=18 inches
Volume of the cone $= \frac{1}{3} \pi r^{2} h$
$= \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times 18 = 924$ cubic inches.
4

If the volume of the cone is 100 cubic units, and height is 12 units, then what will be the lateral height?

10 units
12 units
13 units
5 units
CorrectIncorrect
Correct answer is: 13 units
Volume of the cone $= 100\pi$ cubic units
Height of the cone (h) = 12 units
$\frac{1}{3} r^{2} h = 100\pi$
$\frac{1}{3} \times r^{2} \times 12 =100$
$4r^{2} = 100$
$r^{2} = \frac{100}{4} = 25$
$r = 5$ units
$l = \sqrt{r^{2} + h^{2}}$
$l = \sqrt{5^{2} + 12^{2}} = \sqrt{25 + 144}$
$l = \sqrt{169} = 13$units
5

What is the volume of the right cone for the given radius 6 inches and height 14 inches? $( \pi = 3.14)$

507.52 cubic inches
527.52 cubic inches
537.29 cubic inches
507.51 cubic inches
CorrectIncorrect
Correct answer is: 527.52 cubic inches
Base radius of the cone (r) = 6 inches
Height of the cone (h) = 14 inches
Volume of the cone $= \frac{1}{3} \pi r^{2} h$
$V =\frac{1}{3} \times 3.14 \times 6 \times 6 \times 14 = 527.52$ cubic inches.

Frequently Asked Questions about Volume of a Right Circular Cone

The formula for curved surface area of a right circular cone is $\pi r l$, where r is the radius of the base and l is the slant height.

The formula for total surface area of a right circular cone is $\pi r l + r^{2} = r(l + r)$ where r is the radius of the base and l is the slant height.

If the axis is not perpendicular to the base, the cone is called an oblique circular cone. In the oblique cone, the apex is not aligned with the center of the circular base

Original volume $= \frac{1}{3} \pi r^{2} h$

New Volume $= \frac{1}{3} \times (2r)^{2} \times (2h) = 8 \times \frac{1}{3} \pi r^{2}h = 8$ times original volume

The volume gets 8 times the original volume.

To find the volume of a layer within a right circular cone, use the formula for the volume of a cone $(V = \frac{1}{3} \pi r^{2} h)$, where h and r represent the height and radius of the layer.