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# Surface Area of a Cone – Explanation & Examples

Cone is another important figure in geometry. To recall, a cone is a three-dimensional structure having a circular base where a set of line segments, connecting all of the points on the base to a common point called the apex. It is shown in the figure below.

The vertical distance from the base center to the apex of a cone is the height (h), while the slant height of a cone is the length (l).

**The surface area of a cone is the sum of the area of the slanted, curved surface and area of the circular base. **

In this article, we will discuss **how to find the surface area by using surface area of a cone formula**. We will also discuss the lateral surface area of a cone.

## How to Find the Surface Area of a Cone?

To find the surface area of a cone, you need to calculate the cone’s base and the lateral surface area.

Since the base of a cone is a circle, then the base area (B) of a cone is given as:

**Base area of a cone, B = πr²**

Where ** r** = the base radius of the cone

## Lateral Surface Area of a Cone

The **curved surface of a cone** can be viewed as a triangle whose base length is equal to **2 πr **(circumference of a circle), and its height is equal to the slanted height (

**l**) of the cone.

Since we know, the area of a triangle = ½ bh

Therefore, the lateral surface area of a cone is given as:

Lateral surface area = 1/2×l×2πr

By simplifying the equation, we get,

**The lateral surface area of a cone, (LSA) = πrl**

### Surface area of a cone formula

The total surface area of a cone = Base area + latera surface area. Therefore, the formula for the total surface area of a cone is represented as:

The total surface area of a cone = πr^{2} + πrl

By taking **πr** as a common factor from RHS, we get;

**Total surface area of a cone = πr (l + r) **………………… (Surface area of a cone formula)

Where r = radius of the base and l = slant height

By Pythagorean Theorem, the slant height, **l = √ (h ^{2 }+ r^{2})**

**Solved Examples**

*Example 1*

The radius and height of a cone are 9 cm and 15 cm, respectively. Find the total surface area of the cone.

__Solution__

Given:

Radius, r = 9 cm

Height, h = 15 cm

Slant height, l = √ (h^{2 }+ r^{2})

l = √ (15^{2 }+ 9^{2})

= √ (225 + 81)

=√306

= 17.5

Thus, slant height, l = 17.5 cm

Now substitute the values into the surface area of a cone formula

TSA = πr (l + r)

= 3.14 x 9 (9 + 17.5)

= 28.26 x 157.5

= 4,450.95 cm^{2}

*Example 2*

Calculate the lateral surface area of a cone whose radius is 5 m and slant height is 20 m.

__Solution__

Given;

Radius, r = 5 m

Slant height, l = 20 m

But, the lateral surface area of a cone = πrl

= 3.14 x 5 x 20

= 314 m^{2}

*Example 3*

The total surface area of a cone is 83.2 ft^{2}. If the slant height of the cone is 5.83 ft, find the radius of the cone.

__Solution__

Given;

TSA = 83.2 ft^{2}

Slant height, l = 5 .83ft

But, TSA = πr (l + r)

83.2 = 3.14 x r (5.83 + r)

83.2 = 3.14 x r (5.83 + r)

By applying the distributive property of multiplication on the RHS, we get

83.2 = 18.3062r + 2.14r^{2}

Divide each term by 3.14

26.5 = 3.14r + r^{2}

r^{2 }+ 3.14r – 26.5 = 0

r = 3.8

Therefore, the radius of the cone is 3.8 ft

*Example 4*

The total surface area of a cone is 625 in^{2}. If the slant height is thrice the radius of the cone, find the dimensions of the cone.

__Solution__

Given;

TSA = 625 in^{2}

Slant height = 3 x radius of the cone

Let the radius of the cone be x

Slant height =3x

TSA = πr (l + r)

625 = 3.14x (3x + x)

Divide both sides by 3.14.

199.04 = x(4x)

199.04 = 4x^{2}

Divide both sides by 4 to get

49.76 = x^{2}

x = √49.76

x = 7.05

Therefore, the dimensions of the cone are as follows;

Radius of the cone = 7.05 in

Slant height, l = 3 x 7.05 = 21.15 in

Height of the one, h = √ (21.15^{2} – 7.05^{2})

h = 19.94 in.

*Example 5*

The lateral surface area is 177 cm^{2} less than the total surface area of a cone. Find the radius of the cone.

__Solution__

The total surface area of a cone = Lateral surface area + Base area

Therefore, 177 cm^{2} = Base area

But, the base area of a cone = πr^{2}

177 = 3.14r^{2}

r^{2} = 56.4 cm

r = √56.4

= 7.5 cm

So, the radius of the cone is 7.5 cm.

*Example 6*

The cost of painting a conical container is $0.01 per cm^{2}. Find the total cost of painting 15 conical containers of radius 5 cm and slant height 8 cm.

__Solution__

TSA = πr (l + r)

=3.14 x 5 (5 + 8)

= 15.7 x 13

= 204.1 cm^{2}

The total cost of painting 15 containers = 204.1 x 0.01 x 15

= $30.62