Surface Area of Cone – Formula, Derivation, Examples

What is a cone?

We’re sure you might’ve seen birthday caps and ice cream cones; these are 3-D structures having a circular base and a pointed end (apex). This shape is known as a cone. Since the base is circular, it is evident that it has a radius (r) and a diameter. The axis connecting the centre of the circular base to the cone’s apex is known as the height (h) of the cone. Thus, a cone has three parts – radius, height and slant height. 

A cone is of two types – a right circular cone and an oblique cone. In this article, we will make you more aware of a right circular cone.

Right circular cone

A cone with an axis running perpendicular to the base is called a right circular cone. In this article we will only talk about a right circular cone. The length of the apex and any point on the circumference of the cone is called its slant height (l). All right circular cones are cones, but all cones might not be right circular in nature.

In right circular cones, the radius, height and slant height make a right angled triangle. It means that if we don’t have the value of slant height in the question but have radius and height – we can still calculate the slant height and then calculate the CSA and TSA.

The surface area of a cone

The measure of a cone’s surface area is the area occupied by a cone’s surface. Now, we must see that a cone has two kinds of surface areas:

• Curved surface area
• Total surface area. 

The main difference between the curved surface area and the total surface area (TSA) of a cone is that the TSA consists of the lateral surface area of the cone and the area of the circular base. In contrast, CSA consists of the area of the curved surface, excluding the flat surface. 

The surface area of a cone formula

You can calculate the  CSA and TSA of a cone, by finding the area of the cone’s base and the curved surface area of the cone.

The curved surface area of a cone

You can look at the curved surface of a cone as a triangle. For this, you can slice up the curved surface into thin pieces to approximate them as small triangles. The total base length of these triangles is equal to the circumference of the cone’s circular base, and the slant height of the cone is now the height of each triangle.

The sum of bases of all triangles = circumference of the base of the cone = 2πr

Height of each triangle = slant height of the cone = l

Let us consider the bases of the small triangles to be

b_1,b_2,b_3, ….. respectively.

The heights of all the triangles are equal (l). 

The total area of these triangles will be the curved surface area.

Therefore the CSA = (1/2 \times b_1 \times l) + (1/2 \times b_2 \times l) + ....... 

                                  = (1/2 \times l)  (b_1+ b_2 +b_3+.....)

                                  = (1/2 \times l) \times (2\pi r)

                                  = \pi rl

Therefore, the curved surface area of cone = \pi rl

The total surface area of a cone

The total surface area of a cone = area of circular base + CSA of cone

We know that the area of circular base =\pi r^{2}

And, CSA of cone = \pi rl

So, the total surface area of a cone =\pi r^{2}+ \pi rl

Therefore, the total surface area of a cone = \pi r(r+l)

Also, the slant height of the cone (l)=\sqrt{\left(r^{2}+h^{2}\right)}

Solved examples

Example 1: 

The radius and height of a right circular cone is 4 cm and 16 cm, respectively. What will the total surface area of the cone be?

Solution 

Given, radius (r) = 4 cm

Height of the cone (h) = 16 cm

Slant height of the cone (l)=\sqrt{\left(r^{2}+h^{2}\right)}

⇒ l=\sqrt{\left(16^{2}+4^{2}\right)}

⇒ l=\sqrt{256+16}

⇒ l=\sqrt{272}

∴ l=16.49 cm

TSA of the cone

= \pi r(r+l)

= 3.14 x 4 (4 + 16)

= 3.14 x 80

= 251.2 cm²

Therefore, the TSA of the cone is 251.2 cm² .

Example 2:

The slant height of a cone is 24 cm, and the radius is 10 cm. Find the lateral surface area of the cone?

Solution:

Given, 

Radius (r) = 10 cm

Slant height (l) = 24 cm

Putting these values in the formula, CSA of cone

= \pi rl

=3.14 \times 10 \times 24 \mathrm{~cm}^{2}

=753.6 \mathrm{~cm}^{2}

Therefore, the CSA of the cone is =753.6 \mathrm{~cm}^{2}.