Surface Area & Volume of Right Circular Cone | Mathematics (Maths) Class 9 PDF Download

Surface Area of a Cone

Cone is defined as a three-dimensional solid structure that has a circular base. A cone can be viewed as a set of non-congruent circular disks that are placed over one another such that the ratio of the radius of adjacent disks remains constant. In this article, let us discuss the formula for the surface area of a cone, its derivation with many solved examples.

Surface Area of a cone Formula
The surface area of a cone is the total area occupied by its surface in a 3D plane. The total surface area will be equal to the sum of its curved surface area and circular base area.

Surface area of cone = πr(r + √(h² + r²))
where r is the radius of the circular base
h is the height of cone Or
Surface area of cone = πr(r + L)
where L is the slant height of the cone
And
Curved Surface area of cone = πrl

Derivation of the Surface area of cone
Consider a cone as a triangle that is being rotated about one of its vertices. Now, think of a scenario where we need to paint the faces of a conical flask. Before painting, we need to know the quantity that is required to paint all the walls. We need to know the area of every face of the container to determine the quantity of paint required and this is known as the total surface area. The total surface area of a cone is the sum of areas of its faces. Let’s derive a general Cone formula to calculate the surface area of a cone.
Take a paper cone and cut it along its slant height to observe the figure being formed by the surface of the cone. Mark the two endpoints as A and B and the point of the intersection of lines as O.
If you further cut this figure into multiple pieces’ viz. Ob₁, Ob₂, Ob₃, …….., Ob_n each measuring the same length as the slant height of the original cone, you will observe n triangles are formed out of it.
Now, if you try calculating the total area of this figure, you just need to add an area of these individual triangles.
Hence,
Area of figure = (1 / 2) × (b₁ + b₂ + b₃ + ………….. + b_n) =  (1 / 2) × (length of an entire curved boundary)
Length of entire curved boundary = circumference of base = 2π × r (where r is the radius of the base)
Thus, area of figure = 1 / 2 x 2π × r × l = πrl
Hence, curved surface area of a cone = πrl
Total Surface Area of a Cone
The Total surface area of a cone includes the curved surface as well as area of its base, which is given as
Base Area = πr²
Curved Surface Area = πrl
Thus the total surface area is given by
πr² + πrl = πr(r + l)

Surface area of Right Circular Cone
A circular cone is the one with the circular right section. A right circular cone is a circular cone whose axis is perpendicular to the base.
Surface area of cone = πr(r + √(h² + r²))
where r is the radius of the circular base
h is the height of the cone
Slant height of the cone, L = √(h² + r²)
Therefore,
Surface area = πr (r + L)

Curved Surface Area of a Cone
The curved surface of a cone is the area of the cone excluding the base. In other words, it is the area of the cone when it is unfolded as shown in the above figure as an unrolled lateral area. The formula to calculate the curved surface area of a cone is given by:
Curved Surface Area (CSA) = πrl
Here,
r = Radius of the circular base of the cone
l = Slant height of the cone
When the cone is in the form of an unrolled lateral area, r is referred to as the length of the arc of a sector and l is referred to as the radius of the sector.

Total Surface Area of a Cone
The total surface area of a cone is defined as the total area of the cone occupied in a three-dimensional area. It is equal to the sum of the curved surface and the base of the cone. The formula to calculate the total surface area of a cone is given by:
Total Surface Area (TSA) = CSA + Area of Circular Base
TSA = πr(r + l)

Solved Examples
Q.1. Determine the curved surface area of a cone whose base radius is 7 cm and slant height is 15 cm.
Solution: Curved surface area of a cone = πrl
= (22 / 7)× 7 ×15
=  330 cm²

Q.2. Calculate the Total surface area of a cone whose radius is 8 cm and height is 12 cm.
Solution: We know that the total surface area is given as
πr(r + l)
Also,

Also,

Thus, Total Surface Area = π(8)(8 + 14.42)
= π(8)(22.42)
= 179.36π

Volume of a Cone

The volume of a cone defines the space or the capacity of the cone. A cone is a three-dimensional geometric shape having a circular base that tapers from a flat base to a point called apex or vertex. A cone is formed by a set of line segments, half-lines or lines connecting a common point, the apex, to all the points on a base that is in a plane that does not contain the apex.
A cone can be seen as a set of non-congruent circular disks that are stacked on one another such that the ratio of the radius of adjacent disks remains constant.

Volume of a Cone Formula
In general, a cone is a pyramid with a circular cross-section. A right cone is a cone with its vertex above the center of the base. It is also called right circular cone. You can easily find out the volume of a cone if you have the measurements of its height and radius and put it into a formula.

Derivation of Cone Volume

Therefore, the volume of a cone formula is given as
The volume of a cone = (1 / 3) πr²h cubic units
Where,
‘r’ is the base radius of the cone
‘l’ is the slant height of a cone
‘h’ is the height  of the cone
As we can see from the above cone formula, the capacity of a cone is one-third of the capacity of the cylinder. That means if we take 1 / 3rd of the volume of the cylinder, we get the formula for cone volume.

Note: The formula for the volume of a regular cone or right circular cone and the oblique cone is the same.

Derivation of Cone Volume
You can think of a cone as a triangle which is being rotated about one of its vertices. Now, think of a scenario where we need to calculate the amount of water that can be accommodated in a conical flask. In other words, calculate the capacity of this flask. The capacity of a conical flask is basically equal to the volume of the cone involved. Thus, the volume of a three-dimensional shape is equal to the amount of space occupied by that shape. Let us perform an activity to calculate the volume of a cone.

Take a cylindrical container and a conical flask of the same height and same base radius. Add water to the conical flask such that it is filled to the brim. Start adding this water to the cylindrical container you took. You will notice it doesn’t fill up the container fully. Try repeating this experiment for once more, you will still observe some vacant space in the container. Repeat this experiment once again; you will notice this time the cylindrical container is completely filled. Thus, the volume of a cone is equal to one-third of the volume of a cylinder having the same base radius and height.
Now let us derive its formula. Suppose a cone has a circular base with radius ‘r’ and its height is ‘h’.  The volume of this cone will be equal to one-third of the product of the area of the base and its height. Therefore,
V = 1 / 3 x Area of Circular Base x Height of the Cone
Since, we know by the formula of area of the circle, the base of the cone has an area (say B) equals to;
B = πr²
Hence, substituting this value we get;
V = 1 / 3 x πr² x h
Where V is the volume, r is the radius and h is the height.

Solved Examples
Q.1. Calculate the volume if r= 2 cm and h= 5 cm.
Solution: Given:
r = 2
h = 5
Using the Volume of Cone formula
The volume of a cone = (1 / 3) πr²h cubic units
V = (1 / 3) × 3.14 × 22 × 5
V = (1 / 3) × 3.14 × 4 × 5
V = (1 / 3) × 3.14 × 20
V = 20.93 cm³
Therefore, the volume of a cone = 20.93 cubic units.

Q.2. If the height of a given cone is 7 cm and the diameter of the circular base is 6 cm. Then find its volume.
Solution: Diameter of the circular base = 6 cm.
So, radius = 6 / 2 = 3 cm
Height = 7 cm
By the formula of cone volume, we know;
V = 1 / 3 πr²h
So by putting the values of r and h, we get;
V = 1 / 3 π 3²7
Since π = 22 / 7
Therefore,
V = 1 / 3 x 22 / 7 x 3² x 7
V = 66 cu.cm.