102 LearnersLast updated on July 30th, 2025
Surface Area of Shapes
Understanding the surface area of various geometric shapes is crucial for solving problems related to geometry. The surface area of a shape is the total area covered by its outer surfaces. Different shapes have different formulas for calculating their surface area depending on their properties. In this article, we will explore the surface area of shapes, focusing on cones as an example.
What is the Surface Area of a Shape?
The surface area of a shape is the total area occupied by the boundary or surface of a given shape, measured in square units. Shapes can be 2-dimensional or 3-dimensional, with 3D shapes having both curved and flat surfaces.
For instance, a cone is a 3D shape made by rotating a triangle around one of its sides, resulting in a circular base and a pointed vertex.
Cones have a curved surface and a flat base, leading to two distinct surface areas: the curved surface area and the total surface area.
Cones can be classified as right circular cones or oblique cones, based on the position of the vertex relative to the base.
Surface Area of a Shape Formula
The surface area of a shape can vary based on its dimensions and type. For example, a cone has a curved surface and two types of surface areas: the curved surface area and the total surface area.
Consider a cone with a specific surface area, height (h), slant height (l), and radius (r). A cone's surface areas include: Curved Surface Area of a Cone Total Surface Area of a Cone
Curved Surface Area of a Cone
The curved surface area of a cone is the area of the cone's curved part, excluding its base. This area is also known as the lateral surface area of the cone.
The formula for the curved surface area (CSA) of the cone is given as:
Curved Surface Area = 𝜋rl square units Here, 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 includes the area of both the curved surface and the circular base.
The formula for calculating the total surface area of a cone is:
Total Surface Area = 𝜋r(r + l) square units Where r is the radius of the base, and l is the slant height.
To derive the total surface area of a cone, consider slicing it from the tip to the edge of the base and then unfolding it. This reveals that the curved surface becomes a sector of a circle. For a cone with height (h), radius (r), and slant height (l):
Total surface area of a cone = base area of a cone + curved surface area of a cone Base area of a cone = 𝜋r²
Curved surface area of a cone = 𝜋rl
Substituting these formulas into the total surface area, we get: Total surface area of a cone, T = 𝜋r² + 𝜋rl
Factoring out common terms: T = 𝜋r(r + l)
Volume of a Cone
The volume of a cone indicates how much space is inside it. It is one-third of the volume of a cylinder with the same height and base. The formula for the volume of a cone is: Volume = ⅓(𝜋r²h) cubic units
Confusion between CSA and TSA
Students sometimes assume that the curved surface area (CSA) and the total surface area (TSA) of a cone are the same. This confusion arises because both involve the slant height and the radius. Always remember that CSA refers only to the curved side of the cone, whereas TSA includes both the curved surface and the base.
Mistake 1
Using height instead of slant height
Some students mistakenly use the vertical height (h) instead of the slant height (l) when calculating the curved surface area. Remember that the formula for the curved surface area is πrl, so always use the slant height.
Mistake 2
Using the wrong value for 𝜋
A common mistake is using 22 as the value of 𝜋. Instead, use accurate values like 22/7 or 3.14. 𝜋 is a fundamental constant in mathematics, and using the correct value is crucial for obtaining accurate results.
Mistake 3
Forgetting to include the base area in the total surface area
Students often calculate only the curved surface area and forget to add the area of the circular base. Always include both parts when calculating the total surface area: Total surface area = curved surface area + base area.
Mistake 4
Assuming curved surface area and lateral surface area are different
Some students mistakenly believe that the curved surface area and the lateral surface area are different, but for a cone, they mean the same thing. Therefore, use the same formula for both.
Mistake 5
Solved Examples of Surface Area of Cone
Find the curved surface area of a cone with a radius of 6 cm and a slant height of 12 cm.
CSA = 226.08 cm²
Problem 1
Given r = 6 cm, l = 12 cm. Use the formula: CSA = πrl = 3.14 × 6 × 12 = 226.08 cm²
Find the total surface area of a cone with a radius of 4 cm and a slant height of 9 cm.
Explanation
TSA = 163.28 cm²
Problem 2
Use the formula: TSA = 𝜋r(r + l) = 3.14 × 4 (4 + 9) = 3.14 × 4 × 13 = 3.14 × 52 = 163.28 cm²
A cone has a radius of 7 cm and a height of 10 cm. Find the total surface area.
Explanation
TSA = 280.02 cm²
Problem 3
Find the slant height using: l = √(r² + h²) = √(7² + 10²) = √(49 + 100) = √149 ≈ 12.21 cm Use the TSA formula: TSA = 𝜋r(r + l) = 3.14 × 7 × (7 + 12.21) = 3.14 × 7 × 19.21 = 280.02 cm²
Find the curved surface area of a cone with a radius of 5 cm and a slant height of 8 cm.
Explanation
CSA = 125.6 cm²
Problem 4
CSA = 𝜋rl = 3.14 × 5 × 8 = 125.6 cm²
The slant height of a cone is 20 cm, and its curved surface area is 628 cm². Find the radius.
Explanation
Radius = 10 cm
It is the total area that covers the outside of a shape, including all its surfaces.
1.What are the two types of surface area in a cone?
2.What is the difference between slant height and height?
3.Is curved surface area the same as lateral surface area?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Cone
Students often make mistakes when calculating the surface area of a cone, leading to incorrect answers. Below are some common mistakes and ways to avoid them.
Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
