3 Ways to Find Surface Area of Cones

Surface area of cones can be calculated in three ways i.e., total surface area of cones, curved surface area of cones, and lateral surface areas of cones.

In this article, we will explore all three ways to find the surface area of cones along with the basics of cones, and surface area of cone formulas. We will also solve some examples related to the surface area of cones.

What is Surface Area of a Cone?

Surface area of a cone is the area bounded by the surface of the cone. The surface area of a cone can be determined in 3 ways:

• Total Surface Area
• Lateral Surface Area with Slant Height and Radius of a Cone
• Lateral Surface Area with Height and Radius of a Cone

Surface Area Type	Formula	Units
Curved Surface Area (S)	​ πr√(r2 + h2)	square units
Total Surface Area (T)	πr2 + πr√(r2 + h2)	square units

Cone Diagram

The below diagram represents a cone.

3 Ways to Find Surface Area of Cones

The three ways to find the surface area of cones are:

• Lateral Surface Area with Radius and Height of Cone

• Total Surface Area of Cone

• Lateral Surface Area with Radius and Slant Height of Cone

Curved Surface Area in Terms of Height and Radius of Cone

Curved surface area of a cone is given by the formula:

CSA of Cone = πr√(r² + h²)

where,

• r is Radius of Base of Cone
• h is Height of Cone

Total Surface Area

Total surface area of a cone is given by the formula:

TSA of Cone = πrl + πr²

where

• r is Radius of Base of Cone
• l is Slant Height of Cone

Lateral Surface Area in Terms of Slant Height and Radius of Cone

Lateral surface area of a cone is given by the formula:

LSA of Cone = πrl

where

• r is Radius of Base of Cone
• l is Slant Height of Cone

Examples on Surface Area of Cones

Example 1: Find the surface area of the cone given the height and radius of cone is 12 units and 5 units respectively.

Solution:

Surface area of cone with height and radius is given by:

CSA of cone = πr√(r² + h²)

= π(5)√(5² + 12²)

= 5π√(25 + 144)

= 5π√169

CSA of cone = 65π sq.units

Example 2: Find the total surface area of the cone if the slant height of cone is 10 units and radius of cone is 7 units.

Solution:

Total surface area of the cone is given by:

TSA of cone = πrl + πr²

= π × 7 × 10 + π × 7²

= 70π + 49π

TSA of cone = 119π sq. units

Example 3: Given the slant height of the cone is 12 units and radius of cone is 3 units then, find the lateral surface area of the cone.

Solution:

Lateral surface area with slant height and radius of cone is given by:

LSA of cone = πrl

= π × 12 × 3

LSA of cone = 36π sq. units

Example 4: Find the total surface area of a cone with radius 8 cm and height 15 cm.

Solution:

Total Surface Area = Lateral Surface Area + Base Area

Lateral Surface Area = π × radius × slant height

Lateral Surface Area = π × 8 cm × 17 cm

= 136π square cm

Base Area = π × radius²

= π × (8 cm)²

= 64π square cm

Total Surface Area = 136π + 64π

= 200π square cm

≈ 628.3 square cm

Example 5: A cone has a total surface area of 154 square units and a base radius of 7 units. What is its slant height?

Solution:

We first need to find the lateral surface area and then use it to find the slant height.

Lateral Surface Area = Total Surface Area – Base Area

Base Area = π × radius²

= π × (7 units)²

= 49π square units

Lateral Surface Area = 154 square units – 49π square units

= (154 – 49π) square units

Now, we know Lateral Surface Area = π × radius × slant height

So, 154 – 49π = π × 7 units × slant height

Solving for slant height:

slant height = (154 – 49π)/(7π)

≈ 6.617 units

Practice Questions on Surface Area of Cone

Q1: Find the surface area of the cone given the height and radius of cone is 17 units and 8 units respectively.

Q2: Find the total surface area of the cone if the slant height of cone is 19 units and radius of cone is 10 units.

Q3: Given the slant height of the cone is 24 units and radius of cone is 9 units then, find the lateral surface area of the cone.

Q4: Calculate the total surface area of a cone with radius 4 meters and height 9 meters.

Q5: If the slant height of a cone is 10 inches and the radius of the base is 6 inches, what is its lateral surface area?

FAQs on Surface Area of Cones

What is the Surface Area of Cone?

Area covered by all the surfaces of the cone is called the surface area of cone.

How to Find Surface Area of Cone?

To find the surface area of cone we use following formulas:

• Total Surface Area of Cone = πrl + πr²
• Lateral Surface Area of Cone = πrl
• Curved Surface Area of Cone = πr√(r² + h²)

What is the Curved Surface Area of Right Circular Cone?

Curved surface area of right circular cone is given by:

CSA of cone = πr√(r² + h²)