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.
Table of Content
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
Also, Check:
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)