**Surface
Area of a Cone**

A right circular **cone **is a
solid shape (3D shape) with one circular face and a curved face. It has one
circular edge and one vertex. The circular face is called the base of the cone.
The radius of the cone is called the **base radius**. The perpendicular
distance from the vertex to the centre of the circular base is called the **height
**of the cone.

The real-life examples of the cone
include ice-cream cone, joker’s cap, pylon, etc.

A cone has two surfaces, one is curved
and other is circular plane. The area of the curved surface is called the **curved surface area** or lateral surface area and the area of curved surface and the circular place
surface together is called the total surface area of the cone.

Take a cone made up of paper. Now, cut
the cone along its slant height and spread it on the ground as shown in the
given figure. Let the slant height of the cone be *l*.

Now, divide
the figure in smaller triangles, such as Ob_{1}b_{2}, Ob_{2}b_{3},
and so on.

The area of
all these triangles gives the curved surface area of the cone.

Thus, curved
surface area of the cone = Area of triangles (Ob_{1}b_{2} + Ob_{2}b_{3}
+ …….)

= ½ × b_{1}b_{2} × *l* + ½ × b_{2}b_{3} × *l* + ……….

= ½ × (b_{1}b_{2} + b_{2}b_{3} + ……….)
× *l*

But, b_{1}b_{2}
+ b_{2}b_{3} + ………. = Circumference of the circle = 2Ï€r

Therefore,
the curved surface area of the cone = ½ × 2Ï€r × *l*

* *= Ï€r*l*

**Curved surface area of a cone = ****Ï€****r l**

Where *l*
is the slant height of the cone.

We can
calculate the slant height of the cone if its height and base radius are given.

Using Pythagoras
theorem, l^{2} = h^{2} + r^{2}

Or, l = √(h^{2} + r^{2})

**Hence, the curved surface area of a cone = ****Ï€****r ****×**** ****√****(h ^{2} + r^{2})**

Now, the
total surface area of the cone = Curved surface area + Area of the circular
base

= Ï€r*l*
+ Ï€r^{2}

= Ï€r(*l*
+ r)

**Total surface area of a cone = ****Ï€****r( l + r)**

**Surface
Area of a Cone Formula**

**Curved surface area of a cone = ****Ï€****r l**

**Total surface area of a cone = ****Ï€****r( l + r)**

**Surface
Area of a Cone Example**

**Example
1: **Find the curved
surface area of a cone whose base radius is 14 cm and slant height is 10 cm.

**Solution:** Given: r = 14 cm and l = 10 cm

Curved
surface area of a cone = Ï€r*l*

= 22/7 × 14 × 10

= 440 sq. cm

**Example
2:** Find the total
surface area of a cone whose base diameter is 14 cm and height is 8 cm.

**Solution:** Given: Diameter = 14 cm, then, r = 14/2
= 7 cm

Height, h = 8

l^{2}
= h^{2} + r^{2}

l^{2}
= (8)^{2} + (7)^{2}

l^{2}
= 64 + 49 = 113

l = √113

l = 10.63 cm

Curved
surface area of a cone = Ï€r*l*

= 22/7 × 7 × 10.63

= 233.86 sq. cm

**Example 3:** Find the height of a cone whose base
radius is 21 cm and curved surface area is 1650 sq. cm.

**Solution:** Given: r = 21 cm and curved surface
area of = 1650 sq. cm

Curved
surface area of a cone = Ï€r*l*

1650 = 22/7 × 21 × *l*

1650 = 66
× *l*

*l* = 1650/66

*l* = 25 cm

Using, l^{2}
= h^{2} + r^{2}

h^{2}
= l^{2} – r^{2}

= (25)^{2} – (21)^{2}

= 625 – 441

= 184

h = √184

h = 13.56 cm