# Surface Area of a Cone

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 unitswhere,

“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 unitswhere,

“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 = πr^{2}

⇒ T = πr^{2} + 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) = √(r^{2} + h^{2})

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

Curved surface area (S) = πr√(r^{2}+ h^{2}) square units

Total surface area (T) = πr^{2 }+ πr√(r^{2 }+ h^{2}) 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 is266.9 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) = √(h

^{2}+ r^{2})⇒ h = √(l2 – r2)

= √(25

^{2}– 14^{2}) = √429 = 20.71 unitsThus, 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) = √(h

^{2}+ r^{2})l = √(7

^{2}+ 24^{2}) = √(625) = 25 inchesWe 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?**

**Answer:**

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

Curved surface area = πrlIf 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?**

**Answer:**

Surface area of con can be calculated in two ways,

CSA = πrl

TSA = πrl(r+l)where,

ris radius of cone

lis slant height of cone

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

**Answer:**

Slant height of a cone is defined by the formula

l=√(r^{2}+ h^{2}) 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.**

**Answer:**

Surface of the base of a Cone is circular and the formula for the base of surface of the cone is πr

^{2 }square units.

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

**Answer:**

Surface Area of a Cone is the region occupied by the Surface of a Cone in 3-D space. It can be calculated by finding the sum of the lateral area and the base area of the cone.

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