Volume of a Cone(Formula, Derivation & Examples)

A cone is a 3D shape with a curved surface and a flat circular surface. One of the most common real-world examples of a cone are ice-cream cones are party hats. 

The space occupied by a cone is called the volume of a cone. It’s also referred to as the capacity of a cone and it depends on the radius and the height of the cone. Volume of a cone is measured in terms of cubic units, such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$, etc.

Let’s learn how to find the volume of a cone and its formula.

What is the Volume of a Cone (Volume of Right Circular Cone)?

The volume of a cone is the number of cubic units, occupied by the cone completely. The unit of volume of a cone is cubic units such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$.

volume of a cone

If the measure of radius and height of a cone are $r$ and $h$ unit, then the formula for calculating its volume is $\frac {1}{3}\pi r^{2} h$.

Derivation of Volume of Cone Formula

To derive the formula for the volume of a cone let’s consider a cylinder of radius $r$ and height $h$.

volume of a cone

Now let us locate the centre of the base $O$. From this centre measure a distance to any point on the circumference on the top. This is the slant height of the cone. 

Further, let the circumference of the cylinder be $x$. Now draw a circle with slant height as radius and mark an arc of length $x$. Draw a sector $AOB$ with $L$ as radius and the length of the arc as $x$. And after that cut this sector.

Finally, let’s make cones with this sector. This right circular cylinder and a right circular cone are of the same base radius and the same height.

Fill this cone with sand up to the brim. Pour it into the cylinder. 

Is it filled? No. Repeat it. 

Now is it filled? No, so pour once more. 

Now, the tin is completely filled. So it takes $3$ sand-filled cones to completely fill a $1$ cylinder.

volume of a cone

From this observation, we come to the conclusion that three times the volume of a cone, makes up the volume of a cylinder, which has the same base radius and the same height as the cone, which means that the volume of the cone is one-third the volume of the cylinder.

So, the volume of a cone = $\frac {1}{3} \pi r^{2} h$, where $r$ is the base radius and $h$ is the height of the cone.

Volume of a Cone Examples

Ex 1: Find the volume of a cone of base radius $3.5 in$ and height $12 in$.

The radius of the base of cone $r = 3.5 in$

Height of the cone $h = 12 in$

Volume of cone = $\frac {1}{3} \pi r^{2} h = \frac {1}{3} \times \frac {22}{7} \times 3.5^{2} \times 12 = 154 in^{3}$.

Ex 2: Amount of liquid in a conical flask of base radius $10.5 cm$ is $924 c.c.$. Find the level of water in the flask.

The radius of the base of a conical flask $r = 10.5 cm$

The volume of water in the conical flask $V = 924 c.c. = 924 cm^{3}$

Let the height of the water level in the flask be $h$ cm.

Therefore, $924 = \frac {1}{3} \times \frac{22}{7} \times 10.5^{2} \times h => h = 8 cm$.

Ex 3: If the radius of a cone is doubled and height is halved, then find the ratio of the new volume to the original volume of the cone.

Let the radius of the base of a cone be $r$ and its height is $h$.

Radius is doubled, therefore, a new radius of the cone = $2r$.

Height is halved, therefore, new height of the cone = $\frac {h}{2}$.

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

Volume of the new cone =$\frac {1}{3} \pi \left(2r \right)^{2}\left(\frac {h}{2} \right) = \frac {1}{3} \pi \left(2r^{2}h \right)$.

Therefore, the ratio of new volume to the original volume = $\frac{\frac {1}{3} \pi \left(2r^{2}h \right)}{\frac {1}{3} \pi r^{2}h} = 2:1$.

Famous Mathematicians

Cone – A 3D Solid Shape

A cone is a shape formed by using a set of line segments or lines which connects a common point, called the apex or vertex, to all the points of a circular base(which does not contain the apex). The distance from the cone’s vertex to the base is the height of the cone.

The circular base has a measured value of radius. And the length of the cone from apex to any point on the circumference of the base is the slant height. Based on these quantities, there are formulas derived for the surface area and volume of the cone. In the figure you will see, the cone is defined by its height, the radius of its base, and slant height.

Types of Cone

The cones are broadly divided into two categories.

Right Circular Cone

A cone that has a circular base and the axis from the vertex of the cone towards the base passes through the center of the circular base. The vertex of the cone lies just above the center of the circular base. The word “right” is used here because the axis forms a right angle with the base of the cone or is perpendicular to the base.

These are the most common types of cones that are used in geometry. See the figure below which is an example of a right circular cone.

Oblique Cone

A cone that has a circular base but the axis of the cone is not perpendicular to the base, is called an Oblique cone. The vertex of this cone is not located directly above the centre of the circular base. Therefore, this cone looks like a slanted cone or tilted cone.

volume of a cone

Right Circular Cone

A right circular cone is one whose axis is perpendicular to the plane of the base. We can generate a right cone by revolving a right triangle about one of its legs.

In the figure, you can see a right circular cone, which has a circular base of radius $r$ and whose axis is perpendicular to the base. The line which connects the vertex of the cone to the centre of the base is the height of the cone. The length at the outer edge of the cone, which connects a vertex to the end of the circular base is the slant height.

volume of a cone

Relation Between Radius, Height, and Slant Height of a Right Circular Cone

Since, the radius, height, and slant height of a right circular cone form a right triangle, therefore these three are related to each other by a Pythagoras theorem, where slant height is the hypotenuse and the radius and height are the two legs of the right triangle.

volume of a cone

Therefore, according to the Pythagoras theorem, $l^{2} = r^{2} + h^{2} => l = \sqrt{r^{2} + h^{2}}$

Similarly, $r = \sqrt{l^{2} – h^{2}}$ and $h = \sqrt{l^{2} – r^{2}}$

where $r$ is the radius, $h$ is the height and $l$ is the slant height of a cone.

Conclusion

The volume of a cone is the number of cubic units, occupied by it completely and is calculated by using its radius and height. If $r$ is the radius and $h$ is the height of a cone, then the formula for computing its volume is $\frac {1}{3} \pi r^{2} h$.

Practice Problems

  1. Find the volume of the cone if the height and the slant height of a cone are $18 cm$ and $21 cm$, respectively.
  2. If the radius is $3.5 cm$ and the height is $12 cm$ find the volume of the cone.
  3. The radius and slant height of a cone is $20 cm$ and $29 cm$ respectively. Find its volume.
  4. The height of a cone is $21 cm$. If its volume is $1500 cm^{3}$, find the radius of the base.
  5. If the volume of a cone is $1500 cm^{3}$ and the radius of the base is $7 cm$, find the slant height of the cone.
  6. The circumference of the base of a $12 m$ high wooden solid cone is $44 m$. Find the volume.
  7. If the volume of a cone is $48 \pi cm^{3}$ and the height is $10 cm$, find the radius of its base.

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FAQs

What is the volume of a cone?

The volume of a cone is the amount of space in it. It can be obtained by multiplying one-third of its base area by its height. The volume of a cone of base radius $r$ and height $h$’ is $\frac {1}{3} \pi r^{2}h$.

How do you find the volume of a cone?

The formula used to find the volume of a cone of base radius $r$ and height $h$ is $\frac {1}{3} \pi r^{2} h$.

What is the volume of a cone using diameter?

Let us consider a cone of radius $r$, diameter $d$, and height $h$. The volume of a cone of base radius $r$ and height $h$ is $V = \frac {1}{3} \pi r^{2}h$. 
We also know that $r = \frac {d}{2}$. 
By substituting this in the above formula we get $V = \frac {1}{3} \pi \frac {d^{2}h}{4}$.

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