Volume of a Cone – Formula with Solved Examples

Feb 4, 2022

In this article, Let’s explore the volume of a cone.

The volume of an object is the space occupied by it or the space surrounded by a boundary. For example, let us consider a cone. It is a 3-D geometric shape in which many circular rings form a cone when stacked up. It tapers smoothly from a flat base to the apex or vertex point. Let us find out the volume of a cone through an experiment.

Experiment to Find the Volume of a Cone

Now, we will learn how to find the volume of a cone. We will consider a cylinder and compare its volume to that of a cone to find out the volume of the cone.

Procedure:

Let us take a cone and a cylinder of the same height and base.
Now fill the cone up to the brim with sand and pour this sand into the cylinder.
We can see that it only fills a part of the cylinder.
Again, we will supply the cone with sand and pour it into the cylinder.
We see that the cylinder is still not complete with sand.
We will now put sand in the cone for the third time
Now empty it into the cylinder.
It is clear that the cylinder also fills up to the brim.
It shows that three times the volume of the cone makes up the volume of a cylinder.
Both the cylinder and cone have the same base radius and height.

Hence we can conclude that the volume of the cone is one-third the volume of the cylinder.

Figure (a) shows the cone and cylinder of the same base radius and height. It also depicts the three rounds of filling the cylinder with the sand by emptying the cone.

cone and cylinder

cone and cylinder 2

The Three Main Properties of a Cone are:

It has one circular face.
It has zero edges.
It has one vertex (corner).

Derivation of Cone Volume

Let’s do an activity to determine the volume of a cone.
Take a flask with a conical base and the same height as a cylindrical container. The conical flask should be filled with water before proceeding. Start filling the cylindrical container you took with this water. You’ll see that it doesn’t completely fill the container.

If you try to replicate this experiment, you will still see some empty space in the container. Repeat the experiment a second time, and you’ll see that the cylindrical container is this time totally filled. Thus, the volume of a cylinder with the same base radius and height is equal to one-third of the volume of a cone.

Let’s now calculate its formula. Consider a cone with a height of “h” and a circular base with radius “r.” The area of the base multiplied by the height of the cone will equal one-third of its volume. Therefore,

V = 1/3 x Circular Base Area x Cone Height

The base of the cone has an area (let’s say B) equal to since we know from the formula for the area of the circle;

B = πr2

Thus, when we substitute this value, we obtain;

V = 1/3 x πr2 x h

where V stands for volume, r for radius, and h for height.

Volume of a Cone Formula

The figure below shows a cone with base radius (r), height (h), and slant height (l), giving volume as:

Cone

The volume of a cone = (⅓) πr²h cubic units.

As we can see in the above formula, the capacity of a cone is one-third of the capacity of the cylinder. So that means if we take (⅓)rd of the volume of the cylinder, we get the formula for cone volume.

Volume of a Cone in Real Life

We come across various cone-shaped objects in our daily life. We might then ask ourselves what is the volume of a cone? Or how to find it? The volume of the cone is handy to find the sizes of cone-shaped items. Some of them are:

Traffic cones
Ice-cream cones
Funnels
Christmas Tree
Birthday caps

Generating cones

In this section, we are generating cone solids called pyramids.

Procedure:

Let us cut a right-angle triangle ABC.
Paste a stick on the perpendicular side, say AB of the triangle.
Hold the stick and rotate the triangle about the stick.

Generating cones

It will form a geometric shape called a right circular cone.
Point A is the vertex.
AB is the height (h).
BC is the radius of the base (r).
AC is called the slant height (l) of the cone.

Slant Height

It is the distance from the vertex or apex to the point on the outer line of the cone’s circular base. The Pythagoras Theorem can derive the formula for slant height.

l = √(r²+h²)

Surface Area of the Cone

The surface area of a right circular cone is equal to the sum of its lateral surface area(πrl) and the surface area of the circular base(πr²). Therefore,

The total surface area of the cone = πrl + πr²

Area = πr(l + r)

We can put the value of slant height and calculate the area of the cone.

Types of Cone

There are two types of cones;

Right Circular Cone
Oblique Cone

Right Circular Cone

A cone with a circular base and the axis from the cone’s vertex towards the base passes through the center of the circular base. The cone’s vertex lies just above the center of the circular base as the axis forms a right angle with the base of the cone or is perpendicular to the base.

Oblique Cone

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

Properties of Cone

It has only one face, which is a circular base but no edges
It has only one apex or vertex point.
The volume of a cone is (⅓) πr²h.
The total surface area of the cone is πr(l + r)
The slant height of the cone is √(r²+h²)

Frustum of Right Circular Cone

You might have seen a bucket. Imagine that the bucket is extended in a particular direction. Wouldn’t it meet at a particular point? Which shape does it form? That’s the frustum of a cone! The frustum is that part of the cone left after a plane parallel cuts the cone to its base. For example, in this case, it’s the bucket.