105 LearnersLast updated on August 5th, 2025
Surface Area of Pyramids and Cones
Pyramids and cones are 3-dimensional shapes with distinct geometric properties. The surface area of these shapes is the total area covered by their outer surfaces. For pyramids, it includes the base and the triangular faces, while for cones, it includes both the curved surface and the base. In this article, we will explore the surface area of pyramids and cones.
What is the Surface Area of a Pyramid and a Cone?
The surface area of a pyramid or a cone is the total area occupied by their boundaries or surfaces, measured in square units.
A pyramid consists of a polygonal base and triangular faces converging at a vertex, while a cone has a circular base and a curved surface ending in a vertex.
Both shapes have two surface areas: the lateral surface area (which excludes the base) and the total surface area (which includes the base).
Pyramids are categorized by the shape of their base, such as triangular, square, or rectangular pyramids.
Cones are classified into right circular cones and oblique cones.
A right circular cone has its vertex directly above the center of its base, while an oblique cone's vertex is off-center.
Surface Area Formulas for Pyramids and Cones
Pyramids and cones have different types of surface areas: the lateral surface area and the total surface area. Below are the formulas for each shape, along with their height (h), slant height (l), and base dimensions.
Pyramids:
Lateral Surface Area of a Pyramid
Total Surface Area of a Pyramid
Cones:
Curved Surface Area of a Cone
Total Surface Area of a Cone
Lateral Surface Area of a Pyramid
The lateral surface area of a pyramid is the sum of the areas of its triangular faces, excluding the base. For a pyramid with a regular polygonal base, the lateral surface area can be calculated using:
Lateral Surface Area = (1/2) × Perimeter of Base × Slant Height
Here, the slant height is the height of each triangular face from the base to the vertex of the pyramid.
Total Surface Area of a Pyramid
The total surface area of a pyramid includes the lateral surface area plus the area of the base.
The formula is: Total Surface Area = Lateral Surface Area + Base Area
For a regular pyramid with a base area B and slant height l:
Total Surface Area = (1/2) × Perimeter of Base × Slant Height + Base Area
Curved Surface Area of a Cone
The area of the curved part of a cone, excluding its base, is known as the curved surface area or lateral surface area of a cone. The formula for the CSA (Curved Surface Area) 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 is the sum of the curved surface area and the area of the circular base.
The formula is: Total Surface Area = πr(r + l) square units
Where r is the radius of the base of the cone, and l is the slant height of the cone.
The base area of a cone is calculated as πr².
Therefore, the total surface area is: Total Surface Area = πr² + πrl
Factor out the common terms: TSA = πr(r + l)
Confusion between Lateral Surface Area and Total Surface Area
Students sometimes confuse the lateral surface area (LSA) with the total surface area (TSA). Remember that LSA only includes the slanted sides, while TSA includes the base as well.
Mistake 1
Using Height Instead of Slant Height
Some students mistakenly use the vertical height (h) instead of the slant height (l) when finding the lateral or curved surface area. Always ensure you use the slant height for these calculations.
Mistake 2
Using the Wrong Value for π
A common mistake is using an incorrect value for π. Use accurate values like 3.14 or 22/7 to ensure precise calculations.
Mistake 3
Forgetting to Include the Base Area in Total Surface Area
Students often calculate only the lateral or curved surface and forget to add the base area. Always include both when calculating the total surface area.
Mistake 4
Assuming Curved Surface Area and Lateral Surface Area Are Different
Some students mistakenly believe that curved surface area and lateral surface area are different, but for cones, they mean the same thing. Use the same formula for both.
Mistake 5
Solved Examples of Surface Area of Pyramids and Cones
Find the lateral surface area of a pyramid with a square base of side 6 cm and a slant height of 10 cm.
LSA = 120 cm²
Problem 1
Given base side = 6 cm, l = 10 cm. Perimeter of base = 4 × 6 = 24 cm. Use the formula: LSA = (1/2) × Perimeter of Base × Slant Height = (1/2) × 24 × 10 = 12 × 10 = 120 cm²
Find the total surface area of a cone with radius 5 cm and slant height 13 cm.
Explanation
TSA = 282.6 cm²
Problem 2
Use the formula: TSA = πr(r + l) = 3.14 × 5 × (5 + 13) = 3.14 × 5 × 18 = 3.14 × 90 = 282.6 cm²
A pyramid has a triangular base with a perimeter of 18 cm and a slant height of 8 cm. Find the lateral surface area.
Explanation
LSA = 72 cm²
Problem 3
Given perimeter = 18 cm, l = 8 cm. Use the formula: LSA = (1/2) × Perimeter of Base × Slant Height = (1/2) × 18 × 8 = 9 × 8 = 72 cm²
Find the curved surface area of a cone with radius 4 cm and slant height 7 cm.
Explanation
CSA = 88 cm²
Problem 4
CSA = πrl = (22/7) × 4 × 7 = 22 × 4 = 88 cm²
The slant height of a pyramid is 9 cm, and its lateral surface area is 90 cm². Find the perimeter of its base.
Explanation
Perimeter = 20 cm
It is the total area covering the outside of the pyramid, including its lateral faces 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 Surface Area Calculations for Pyramids and Cones
Students often make mistakes when calculating the surface area of pyramids and cones, which leads to incorrect answers. Below are some common mistakes and ways to avoid them.
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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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