 # Surface Area Of A Cone

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

What is 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.

## Derivation of Surface area of cone

Consider a cone as a triangle which is being rotated about one of its vertexes. 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 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, …….., Obn 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 + ………….. + bn)

=  (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 =  $\frac{1}{2} \times 2 \pi \times r \times l= \pi rl$

Hence, curved surface area of a cone = $\pi rl$

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 = $\pi r^{2}$

Curved Surface Area = $\pi rl$

Thus the total surface area is given by-

$\pi r^{2} + \pi r l = \pi r (r+ l)$

### 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 cone

Slant height of the cone, L = √(h2+r2)

Therefore,

Surface area = πr (r + L)

## Solved Examples

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

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

$\pi r (r+ l)$

Also, $l = \sqrt{h^{2}+ r^{2}}$

Also, $l = \sqrt{8^{2}+ 12^{2}} = \sqrt{208} = 14.42$

Thus Total Surface Area $= \pi (8) (8 + 14.42)$

$= \pi (8) (22.42)$

$= 179.36 \pi$