Volume of a Prism Formula
The volume of a prism is the amount of space it occupies. A prism has a uniform cross-sectional area along its height, which means the volume can be calculated by multiplying the area of the base by its height. This approach works regardless of the shape of the base (such as rectangular, triangular, or polygonal).
Below are the formulas for calculating the volume of a prism, with detailed explanations and examples for each type.
Formulas for the Volume of a Prism
General Formula Using Base Area and Height:
If the area \( A \) of the base and the height \( h \) of the prism are known,
The volume \( V \) can be calculated with the formula:
\( V = A \times h \)
In this formula:
- \( A \) is the area of the base of the prism
- \( h \) is the height of the prism, or the perpendicular distance between the two bases
Depending on the shape of the base, the formula for the area \( A \) will differ. Below are specific formulas for prisms with different base shapes.
Volume 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 \( A = l \times w \). The volume \( V \) of the rectangular prism is then calculated as:
\( V = l \times w \times h \)
Example 1: Calculating Volume of a Rectangular Prism
Problem: Find the volume 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: \( V = l \times w \times h \).
- Substitute \( l = 5 \), \( w = 3 \), and \( h = 10 \): \( V = 5 \times 3 \times 10 \).
- Multiply: \( V = 150 \, \text{cm}^3 \).
The volume of the rectangular prism is \( 150 \, \text{cm}^3 \).
Volume of a Triangular Prism
If the base of the prism is a triangle with base \( b \) and height \( h_b \), the area of the base \( A \) is \( A = \dfrac{1}{2} b \times h_b \). The volume \( V \) of the triangular prism is then calculated as:
\( V = \dfrac{1}{2} b \times h_b \times h \)
Example 2: Calculating Volume of a Triangular Prism
Problem: A triangular prism has a base length \( b = 4 \, \text{cm} \), base height \( h_b = 6 \, \text{cm} \), and prism height \( h = 12 \, \text{cm} \). Find its volume.
Solution:
- Write down the formula: \( V = \dfrac{1}{2} b \times h_b \times h \).
- Substitute \( b = 4 \), \( h_b = 6 \), and \( h = 12 \): \( V = \dfrac{1}{2} \times 4 \times 6 \times 12 \).
- Multiply: \( V = \dfrac{1}{2} \times 288 = 144 \, \text{cm}^3 \).
The volume of the triangular prism is \( 144 \, \text{cm}^3 \).
Volume 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. Once \( A \) is known, the volume \( V \) is calculated as:
\( V = A \times h \)
Example 3: Calculating Volume of a Hexagonal Prism
Problem: A hexagonal prism has a base with side length \( s = 5 \, \text{cm} \) and a height \( h = 15 \, \text{cm} \). The area of a hexagonal base \( A \) is given by \( A = \dfrac{3\sqrt{3}}{2} s^2 \). Find the volume of the hexagonal prism.
Solution:
- Calculate the area of the hexagonal base: \( A = \dfrac{3\sqrt{3}}{2} \times 5^2 \).
- Calculate \( s^2 \): \( A = \dfrac{3\sqrt{3}}{2} \times 25 = 37.5\sqrt{3} \approx 64.95 \, \text{cm}^2 \) (using \( \sqrt{3} \approx 1.732 \)).
- Now use the volume formula: \( V = A \times h \).
- Substitute \( A \approx 64.95 \) and \( h = 15 \): \( V \approx 64.95 \times 15 \).
- Multiply: \( V \approx 974.25 \, \text{cm}^3 \).
The volume of the hexagonal prism is approximately \( 974.25 \, \text{cm}^3 \).
These examples demonstrate how to calculate the volume of a prism with different base shapes, including rectangular, triangular, and polygonal bases, using the general formula \( V = A \times h \).