# Surface Area of a Cone

• Last Updated : 09 Nov, 2022

Surface Area of a Cone is the total area occupied by the surfaces of the cone. A cone is a three-dimensional-shaped geometric figure that has a flat face and a curved surface with a pointed end. The shape of a cone is obtained by stacking several triangles and then rotating them around an axis. The pointed end of the cone is called an apex or a vertex. The perpendicular distance between the center of the base and the apex of a cone is called the height, while the slant height is the distance from the apex of the cone to any point on the circumference of the base. A normal cone is also called a right circular cone. The area of a right circular cone is discussed in this article.  Ice cream cones, traffic cones, funnels, birthday hats, etc. are some examples of cones that we see in our daily life.

## What is the Surface Area of a Cone?

Surface Area of a cone is visualized as the area occupied by the cone when it is cut open. It is formed by a circular base and a curved surface. The surface area of the cone is dependent on the radius of its base and the height of the cone. Also, the volume of a cone depends on its radius and height.

## Surface Area of a Cone Formula

The surface area of a cone is defined as the area occupied by the boundary or the surface of a cone. A cone has two kinds of surface areas namely, a curved surface area and a total surface area.

### Curved Surface Area of a Cone

The curved surface of a cone is defined as the area of the curved part of the cone, i.e., the area of the cone excluding its base.

Curved Surface Area (S) = πrl square units

where,
r” is the radius of the base of a cone and
l” is the slant height of the cone.

### Total Surface Area of a Cone

The total surface area of a cone is defined as the total area occupied by a cone in a three-dimensional space, i.e., the area of the curved surface and the area of the circular base.

Total Surface Area = πr (r + l) square units

where,
r” is the radius of the base of a cone and
l” is the slant height of the cone.

## Derivation of Surface Area of a Cone

To observe the figure formed by the surface of a cone, take a paper cone and then cut it along its slant height. Now, mark A and B as the two endpoints and O as the point of the intersection of the two lines. Now if we open this, it will look like a sector of a circle.

So, to find the curved surface area of the cone, we have to find the area of the sector.

The area of the sector in terms of length of arc = (arc length × radius)/ 2 = ((2πr) × l)/2 = πrl.

Therefore,

Curved Surface Area of a Cone (S) = πrl square units

The total surface area of a cone (T) =  Area of the base + Curved Surface area

Since the base is a circle, the area of the base = πr2

⇒ T = πr2 + πrl  = πr(r + l)

Hence,

Total Surface Area of the Cone = πr (r + l) square units

Also, learn about the frustum of a cone here.

## Relation Between the Surface Area of a Cone and Its Height

We know that, the slant height of a cone (l) = √(r2 + h2)

So, by replacing the value of slant in the surface areas formulae of a cone, we get

Curved surface area (S) = πr√(r2 + h2) square units

Total surface area (T) = πr2 + πr√(r2 + h2) square units

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

Let’s consider an example to see how to find the surface area of a cone using its formula.

Example: Find the total surface area of a cone if its radius is 5 inches and its slant height is 12 inches. (Use π = 3.14)

Step 1: Note down the dimensions of the given cone. Here, the slant height of a cone is 12 inches and its radius is 5 inches.

Step 2: We know that the total surface area of the cone = πr (r + l). So, substitute the value of given dimensions in the equation = (3.14) × 5 × (5 + 12)  =  266.9‬ sq. in.

Step 3: Hence, the total surface area of a cone is 266.9‬ square inches.

## Solved Examples on the Surface Area of Cone

Example 1: Find the total surface area of a cone if its radius is 15 cm and its slant height is 10 cm. (Use π = 3.14)

Solution:

Given data,

The radius of cone (r) = 15 cm

The slant height (l) = 10 cm

We know that,

The total surface area of the cone = πr (r + l) square units

= (3.14) × 15 × (15 + 10)

= 1,177.5‬ sq. cm

Hence, the total surface area of the cone is 1,177.5‬ sq. cm.

Example 2: What is the height of a cone if its radius is 14 units and its curved surface area is 1100 square units? (Use π = 22/7)

Solution:

Given data,

The radius of cone (r) = 14 units

The curved surface area of the cone = 1100 square units

Let the slant height of the cone be “l” and the height of the cone be “h”.

We know that,

The curved surface area of the cone = πrl square units

⇒ 1100 = (22/7) × 14 × l

⇒ 44 × l = 1100

⇒ l = 1100/44 = 25 units

We know that,

slant height (l) = √(h2 + r2)

⇒ h = √(l2 – r2)

= √(252 – 142) = √429 = 20.71 units

Thus, the height of the cone is 20.71 units.

Example 3: Determine the slant height of the cone if the total surface area of the cone is 525 sq. cm and the radius is 7 cm. (Use π = 22/7)

Solution:

Given data,

The radius of cone (r) = 7 cm

The total surface area of the cone = 525 sq. cm

et the slant height of the cone be “l”.

We know that,

The total surface area of the cone = πr (r + l) square units

⇒ (22/7) × 7 × (7 + l) = 525

⇒ 22 × (7 + l) = 525

⇒ 7 + l = 25

⇒ l = 18 cm

Therefore, the slant height of the cone is 18 cm.

Example 4: Calculate the lateral surface area of a cone whose radius is 24 inches and height is 7 inches. (Use π = 3.14)

Solution:

Given data,

The radius of cone (r) = 24 inches

The height of the cone (h) = 7 inches.

We know that,

slant height (l) = √(h2 + r2)

l = √(72 + 242) = √(625) = 25 inches

We know that,

The curved surface area of the cone = πrl square units

= 3.14 × 24 × 25

= 1884 sq. in

Hence, the curved surface area of the cone is 1884 sq. in.

## FAQs on Surface Area of Cone

Question 1: What happens to the curved surface area of a cone when its height is doubled?

Curved Surface area of the cone directly depends on the radius of its base.

Curved surface area = πrl

If the radius of cone is doubled its curved surface area also gets doubled.

CSA = π(2r)(l)

= 2πrl

= 2 × original curved surface area

Question 2: How do you find the surface area of the cone?

Surface area of con can be calculated in two ways,

CSA = πrl

TSA = πrl(r+l)

where,

l is slant height of cone

Question 3: Write the ways to calculate the slant height of a cone.

Slant height of a cone is defined by the formula

l=√(r2 + h2) units,

where “r” is the radius and “h” is the height of a cone.

Question 4: Write the formula for the base surface of a Cone.

Surface of the base of a Cone is circular and the formula for the base of surface of the cone is πr2 square units.

Question 5: What do we mean by the Surface Area of a Cone?