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Last updated on September 17, 2025
The area of a prism refers to the total exterior surface area of the three-dimensional shape. Various formulas are used to calculate the area of different types of prisms. These calculations are vital in fields such as engineering, construction, and design. In this section, we will explore how to find the surface area of a prism.
A prism is a three-dimensional solid with two identical ends or faces and flat rectangular sides.
The area of a prism is the total surface area covering the prism, including its lateral (side) area and the area of its two bases.
To find the surface area of a prism, we use the formula: Surface Area = Lateral Area + 2 × Base Area. The lateral area is calculated as the perimeter of the base times the height of the prism. For each specific type of prism, the base area is found using the appropriate formula for that shape. For example, the base area of a rectangular prism is calculated as length × width.
Derivation of the formula:- The lateral area is calculated by multiplying the perimeter of the base (P) by the height (h) of the prism: Lateral Area = P × h. The total surface area is then the sum of the lateral area and twice the area of the base: Surface Area = P × h + 2 × Base Area.
We can find the surface area of a prism using the following methods, depending on the shape of its base. The general approach involves calculating the lateral area and the areas of the bases. The specific methods are: Method using the rectangular base
Method using the triangular base
Method using the polygonal base
Now let’s discuss the methods mentioned.
Method Using a Rectangular Base For a rectangular prism, the lateral area is found by multiplying the perimeter of the rectangular base by the height of the prism. The base area is simply the product of its length and width. For example, if the base dimensions are 4 cm by 6 cm, and the height is 10 cm, what will be the surface area of the prism? Surface Area = (2 × (4 + 6) × 10) + 2 × (4 × 6) = (2 × 10 × 10) + 2 × 24 = 200 + 48 = 248 cm²
Method Using a Triangular Base For a prism with a triangular base, calculate the lateral area using the perimeter of the triangle and the height of the prism. Then, use the formula for the area of a triangle to determine the base area. For example, if the sides of the triangle are 3 cm, 4 cm, and 5 cm, and the height of the prism is 12 cm, find the surface area. Surface Area = (3 + 4 + 5) × 12 + 2 × (1/2 × 3 × 4) = 12 × 12 + 2 × 6 = 144 + 12 = 156 cm²
We measure the surface area of a prism in square units. The measurement depends on the system used:
In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).
In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).
There are various types of prisms, and the formula for finding the area can vary slightly depending on the shape of the base. Here are some special cases:
Case 1: Rectangular Prism Use the formula: Surface Area = 2 × (lw + lh + wh), where l is the length, w is the width, and h is the height.
Case 2: Triangular Prism Use the formula: Surface Area = Lateral Area + 2 × Base Area, where the lateral area is the perimeter of the base times the height.
Case 3: Polygonal Prism For prisms with polygonal bases, calculate the perimeter of the base, find the base area using the appropriate formula, and then apply the general surface area formula.
To ensure accurate results while calculating the area of a prism, keep these tips in mind:
It is common for students to make errors while finding the area of a prism. Let’s examine some frequent mistakes and how to avoid them.
A rectangular prism has dimensions 8 m by 5 m by 3 m. What is the surface area of the prism?
We will find the surface area as 158 m².
For a rectangular prism, the surface area is calculated as 2 × (lw + lh + wh).
Here, the length (l) is 8 m, the width (w) is 5 m, and the height (h) is 3 m.
Therefore, Surface Area = 2 × (8 × 5 + 8 × 3 + 5 × 3) = 2 × (40 + 24 + 15) = 2 × 79 = 158 m².
What is the surface area of a triangular prism with a base perimeter of 12 cm, a height of the prism of 10 cm, and a base area of 6 cm²?
We will find the surface area as 132 cm².
For a triangular prism, the surface area is calculated as Lateral Area + 2 × Base Area.
Here, the lateral area is the perimeter of the base times the height, so 12 × 10 = 120 cm².
Base Area = 6 cm².
Therefore, Surface Area = 120 + 2 × 6 = 120 + 12 = 132 cm².
A hexagonal prism has a base perimeter of 30 m, a height of 8 m, and a base area of 65 m². What is the surface area?
We find the surface area as 490 m².
The surface area of a hexagonal prism is calculated as Lateral Area + 2 × Base Area.
Here, the lateral area is the perimeter of the base times the height, so 30 × 8 = 240 m².
Base Area = 65 m².
Therefore, Surface Area = 240 + 2 × 65 = 240 + 130 = 370 m².
Find the surface area of a square prism with a side length of 7 cm and a height of 15 cm.
We will find the surface area as 686 cm².
For a square prism, the surface area is calculated as Lateral Area + 2 × Base Area.
The perimeter of the base is 4 × side length = 28 cm, and the base area is side length² = 49 cm².
Lateral Area = 28 × 15 = 420 cm².
Therefore, Surface Area = 420 + 2 × 49 = 420 + 98 = 518 cm².
Help Alice find the surface area of a pentagonal prism with a base perimeter of 25 m and a height of 12 m. The base area is 50 m².
We will find the surface area as 400 m².
The surface area of a pentagonal prism is calculated as Lateral Area + 2 × Base Area.
The lateral area is the perimeter of the base times the height, so 25 × 12 = 300 m².
Base Area = 50 m².
Therefore, Surface Area = 300 + 2 × 50 = 300 + 100 = 400 m².
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.
: She has songs for each table which helps her to remember the tables