Surface Area of a Prism Formula
The surface area of a prism is the total area of all its faces, which include the lateral faces (the sides) and the two bases. The surface area calculation depends on the shape of the prism’s base. Below are the formulas for calculating the surface area of a prism, with detailed explanations and examples for each type of base shape.
Formulas for the Surface Area of a Prism
General Formula Using Base Perimeter and Height:
If the perimeter \( P \) of the base and the height \( h \) of the prism are known,
The lateral surface area (only the sides) \( S_L \) can be calculated with the formula:
\( S_L = P \times h \)
The total surface area \( S \), which includes both bases and the lateral area, is:
\( S = 2A + P \times h \)
In these formulas:
- \( A \) is the area of the base of the prism
- \( P \) is the perimeter of the base of the prism
- \( h \) is the height of the prism, or the perpendicular distance between the two bases
Below are specific formulas for prisms with different base shapes, with examples for each.
Surface Area of a Rectangular Prism
If the base of the prism is a rectangle with length \( l \) and width \( w \), the area of the base \( A \) is \( A = l \times w \), and the perimeter \( P \) of the base is \( P = 2(l + w) \). The total surface area \( S \) of the rectangular prism is then calculated as:
\( S = 2(l \times w) + 2(l + w) \times h \)
Example 1: Calculating Surface Area of a Rectangular Prism
Problem: Find the surface area of a rectangular prism with length \( l = 5 \, \text{cm} \), width \( w = 3 \, \text{cm} \), and height \( h = 10 \, \text{cm} \).
Solution:
- Write down the formula: \( S = 2(l \times w) + 2(l + w) \times h \).
- Substitute \( l = 5 \), \( w = 3 \), and \( h = 10 \): \( S = 2(5 \times 3) + 2(5 + 3) \times 10 \).
- Calculate \( l \times w \): \( S = 2 \times 15 + 2 \times 8 \times 10 \).
- Multiply: \( S = 30 + 160 = 190 \, \text{cm}^2 \).
The surface area of the rectangular prism is \( 190 \, \text{cm}^2 \).
Surface Area of a Triangular Prism
If the base of the prism is a triangle with base \( b \), height \( h_b \), and side lengths \( a \), \( b \), and \( c \), the area of the base \( A \) is \( A = \dfrac{1}{2} b \times h_b \), and the perimeter \( P \) of the base is \( P = a + b + c \). The total surface area \( S \) of the triangular prism is then calculated as:
\( S = 2 \left(\dfrac{1}{2} b \times h_b\right) + (a + b + c) \times h \)
Example 2: Calculating Surface Area of a Triangular Prism
Problem: A triangular prism has a base with side lengths \( a = 3 \, \text{cm} \), \( b = 4 \, \text{cm} \), and \( c = 5 \, \text{cm} \), base height \( h_b = 3 \, \text{cm} \), and prism height \( h = 8 \, \text{cm} \). Find its surface area.
Solution:
- Calculate the area of the triangular base: \( A = \dfrac{1}{2} \times 4 \times 3 = 6 \, \text{cm}^2 \).
- Calculate the perimeter of the base: \( P = a + b + c = 3 + 4 + 5 = 12 \, \text{cm} \).
- Now use the total surface area formula: \( S = 2A + P \times h \).
- Substitute \( A = 6 \), \( P = 12 \), and \( h = 8 \): \( S = 2 \times 6 + 12 \times 8 \).
- Multiply: \( S = 12 + 96 = 108 \, \text{cm}^2 \).
The surface area of the triangular prism is \( 108 \, \text{cm}^2 \).
Surface Area of a Prism with a Polygonal Base
For prisms with polygonal bases (such as pentagonal or hexagonal prisms), the area of the base \( A \) can be calculated using the appropriate formula for that polygon’s area, and the perimeter \( P \) as the sum of the polygon’s side lengths. Once \( A \) and \( P \) are known, the total surface area \( S \) is calculated as:
\( S = 2A + P \times h \)
Example 3: Calculating Surface Area of a Hexagonal Prism
Problem: A hexagonal prism has a base with side length \( s = 4 \, \text{cm} \) and a height \( h = 10 \, \text{cm} \). The area of a hexagonal base \( A \) is given by \( A = \dfrac{3\sqrt{3}}{2} s^2 \), and the perimeter \( P = 6 \times s \). Find the surface area of the hexagonal prism.
Solution:
- Calculate the area of the hexagonal base: \( A = \dfrac{3\sqrt{3}}{2} \times 4^2 = 24\sqrt{3} \approx 41.57 \, \text{cm}^2 \).
- Calculate the perimeter of the base: \( P = 6 \times 4 = 24 \, \text{cm} \).
- Now use the total surface area formula: \( S = 2A + P \times h \).
- Substitute \( A \approx 41.57 \), \( P = 24 \), and \( h = 10 \): \( S \approx 2 \times 41.57 + 24 \times 10 \).
- Multiply: \( S \approx 83.14 + 240 = 323.14 \, \text{cm}^2 \).
The surface area of the hexagonal prism is approximately \( 323.14 \, \text{cm}^2 \).
These examples demonstrate how to calculate the surface area of a prism with different base shapes, including rectangular, triangular, and polygonal bases, using the general formulas for lateral and total surface area.