If you frequently eat ice cream, then you might have observed the shape of the cone. Well, the shape of the cone is a cone, but how much ice cream can come in that cone? Have you observed that? Is there any formula to calculate how much ice cream can accommodate in that cone? And does a big cone have a larger quantity of ice cream than the smaller one? Let’s explore the answer to all these questions on this topic.

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.

The three main properties of a cone are:

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

Formula to find the volume of a cone

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

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.

The volume of cones 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.

• 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

Or

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;

1. Right Circular Cone
2. 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 the 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.

How do you find the volume of a cone? Let’s learn through examples

 Example 1: Find the volume of the cone. If the height and the slant height of a cone are 18 cm and 21 cm, respectively. Solution: We know that  l² = r²+ h² r = √(21² – 18²) r = 10.81 cm Volume of the cone is = (⅓) πr²h = (⅓) x (22/7) x 10.81² x 18  = 2203.57 cm² Example 2: If the radius is 3.5 cm and the height is 12 cm. Find the volume of the cone. Solution: The radius of the circular base = 3.5 cm Height = 12 cm We know that volume of cone is V = (⅓) πr²h Since π = 22/7. Therefore, V = (⅓) x (22/7) x 3.5² x 12 V = 154 cm³ Example 3: The height of a cone is 21 cm. If its volume is 1500 cm³, find the radius of the base (r). Solution: The height of the cone is = 21 cm We know that volume of a cone is V = (⅓) πr²h Since π = 22/7. Therefore, 1500 = (⅓) x (22/7) x r² x 21 r = 8.26 cm Example 4: If the volume of a cone is 1500 cm³ and the radius of the base is 7 cm. Find the slant height (l) of the cone. Solution: The radius of the base is = 7 cm We know that volume of cone is V = (⅓) πr²h Since π = 22/7. Therefore, 1500 = (⅓) x (22/7) x 7² x h h = 29.21 cm We know that,  l² = r² + h² l = √(7² + 29.21²) l = 30.03 cm Example 5: If the volume of a cone is 48π cm³ and the height is 10 cm. Find the radius of its base. Solution: The height of the cone is = 10 cm We know that volume of a cone is V = (⅓) πr²h Since π = 22/7. Therefore, 48π  = (⅓) x π x r² x 10 r = 3.79 cm