What is a Cone?

 

A cone is a solid having a circular base and whose lateral surface tapers into a point. And this line joining the vertex to the circular base is called the axis of the cone. When this axis is perpendicular to the base at the center O of the circular base, we say the cone is a right circular cone. Let's learn how to calculate the volume of a cone but before that look at the definition of the cone.

 

Cone Definition

 

A cone is a three-dimensional geometrical figure with one circular base that connects the base and the vertex (the pointed edge).

 

 

Examples of a Cone

 

1. Ice cream cone
2. Party hat
3. Funnel
4. Traffic Cone
5. Megaphone
6. Pencil Tip

 

Elements of a Cone 

 

 

Let us understand the 3 elements of the cone which are:

 

1. Radius of a cone(r): The radius of the circle at the base is the radius of the cone. 
2. Height of a cone(h): The height of the cone is the perpendicular distance between the circular base and vertex of the cone.
3. Slant Height of a cone(l): The slant height of the cone is the distance from the vertex of the cone or the pointy edge to the circumference of the circular base.

 

Relationship between the slant height, height and radius of the cone is:

 

L = ?(r2+h2)

 

Properties of a Cone

 

1. A cone has a circular base which is its only face 
2. A cone has no edge
3. The volume of the cone is ? ?r2h.
4. The total surface area of the cone is ?r(l + r)
5. The slant height of the cone is ?(r2+h2)

 

Types of Cone 

 

Cone can be categorized into two types:

 

1. Right Circular Cone: When the axis of the cone is perpendicular or forms 90 degrees angle with the circular base of the cone so that the vertex of the cone lies just above the center of the circular base, this type of cone is known as Right Circular Cone.

 

2. Oblique Cone: A type of cone which has a circular base but the axis of the cone is not perpendicular to the base. Oblique Cone can also be called a slanted cone or tilted cone because the vertex of this oblique cone is not located directly above the centre of the circular base. 

 

 

What is the Volume of a Cone?

 

The amount of space covered by the 3-dimensional cone is called the volume of a cone. The volume is measured in cubic units, for example, cm3, m3, etc. 

 

Derivation of the Volume of a Cone

 

For finding the formula of volume of cone we will take help of the cylinder.

 

 

Let us take a cylindrical tin. Now let us locate the centre of the base and name it O. From this center we measure distance to any point on the circumference on the top. This is the slant height of the cone. Let the circumference of the cylinder be x cm. We have to draw a circle with slant height as radius and mark an arc of x cm. Draw a sector AOB with L as radius and length of the arc as x cm . Now cut this sector. 

 

Let us 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 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 1 cylinder.

 

 

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, Volume of a Cone = ? ?r2h. Where r is the base radius and h is the height of the cone

 

Volume of a Cone = ? ?r2h. 

 

Finding Volume of a Cone Examples

 

1. The height and the slant height of a cone are 21 cm and 28 cm respectively. Find the volume of the cone.

 

Solution:

 

 

On solving we have 7546 cm cube.

 

2. Monica has a piece of canvas whose area is 551 m square. She uses it to have a conical tent made (base is not included), with a base radius of 7 m. Assuming that all the stitching margins and the wastage incurred while cutting, amounts to approximately 1 m square, find the volume of the tent that can be made with it.

 

Solution : Since the area of the canvas = 551 m square and area of the canvas lost in wastage is 1 m square, therefore the area of canvas available for making the tent is

 

(551 � 1) m square = 550 m square.

 

Now, the surface area of the tent = 550 m square and the required base radius of the conical tent = 7 m

 

Note that the tent has only a curved surface (the floor of a tent is not covered by canvas).

 

Therefore, the curved surface area of the tent = 550 m square.

 

That is, ?rL = 550

 

?22/7 x 7 x L = 550

 

?22  x L = 550

 

?L = 550 x 22

 

?L = 25 m

 

Read More: Volume of a Sphere: Formula, Sphere Properties, and Solved Examples