104 LearnersLast updated on August 30, 2025
Surface Area of Geometric Shapes
Geometric shapes come in various forms and dimensions, each having a unique surface area. The surface area is the total area covered by the outer surface of a shape. For instance, the surface area of a cone includes its curved surface and base. This article explores the surface area of geometric shapes, focusing on the cone.
What is the Surface Area of a Cone?
The surface area of a cone is the total area occupied by its boundary or surface.
Measured in square units, a cone is a 3D shape formed by rotating a right-angled triangle around one of its sides.
It features a circular base at the bottom and a pointed vertex on top.
The cone’s surface includes a curved side and a flat base, giving rise to two surface areas: the curved surface area and the total surface area.
Cones can be classified as right circular cones, where the vertex aligns directly above the base's center, or oblique cones, where the vertex is offset.
Surface Area of a Cone Formula
A cone's surface area is divided into two types: the curved surface area and the total surface area.
These components are defined by the cone's height (h), slant height (l), and radius (r).
A cone has two types of surface areas:
1. Curved Surface Area of a Cone
2. Total Surface Area of a Cone
Curved Surface Area of a Cone
The curved surface area of a cone, excluding its base, is referred to as the lateral surface area.
The formula for the Curved Surface Area (CSA) of a cone is:
Curved Surface Area = πrl square units Where r is the radius of the base, 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 the curved surface and the base. It is calculated using the formula:
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, imagine slicing the cone from its tip to the base edge and then unfolding it.
The curved surface forms a sector of a circle. For a cone with height (h), base radius (r), and slant height (l):
Total Surface Area = Base Area + Curved Surface Area
Base Area = πr² Curved Surface Area = πrl
Substituting yields: Total Surface Area = πr² + πrl
Factor out common terms: T = πr(r + l) Thus, the total surface area T = πr(r + l).
Volume of a Cone
The volume of a cone indicates its internal space or capacity. It is one-third that of a cylinder with identical height and base. The cone's volume is calculated using: Volume = ⅓(πr²h) cubic units
Confusion between CSA and TSA
Students often confuse the curved surface area (CSA) with the total surface area (TSA) of a cone, as both involve the slant height and radius. Remember, CSA pertains to the curved side only, while TSA includes both the curved surface and the base.
Mistake 1
Using height instead of slant height
A common error is using the vertical height (h) instead of the slant height (l) when calculating the curved surface area. Remember, the formula πrl uses the slant height.
Mistake 2
Using the wrong value for π
Students might incorrectly use 22 as the value for π. Always use precise values like 22/7 or 3.14. π is a crucial constant in mathematics, and accuracy is vital for correct results.
Mistake 3
Forgetting to include the base area in the total surface area
Students may calculate only the curved surface and omit the circular base. Ensure both parts are included when determining 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 believe curved surface area and lateral surface area differ, but in a cone, they are synonymous. 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 radius 4 cm and slant height 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 5 cm and a height of 7 cm. Find the total surface area.
Explanation
TSA = 201.06 cm²
Problem 3
Find the slant height using: l = √(r² + h²) = √(5² + 7²) = √(25 + 49) = √74 ≈ 8.6 cm Use the TSA formula: TSA = πr(r + l) = 3.14 × 5 × (5 + 8.6) = 3.14 × 5 × 13.6 = 3.14 × 68 = 213.52 cm²
Find the curved surface area of a cone with radius 4.5 cm and slant height 7 cm.
Explanation
CSA = 98.91 cm²
Problem 4
CSA = πrl = 3.14 × 4.5 × 7 = 3.14 × 31.5 = 98.91 cm²
The slant height of a cone is 18 cm, and its curved surface area is 792 cm². Find the radius.
Explanation
Radius = 14 cm
It is the total area covering the cone's exterior, including the curved side and the base.
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
Errors often occur when calculating a cone's surface area, leading to incorrect results. Below are common mistakes and how 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.
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