Volume of a Pyramid Formula
The volume of a pyramid is the amount of space it occupies. A pyramid has a polygonal base and triangular faces that meet at a single apex. The volume of a pyramid can be calculated by using the area of the base and its height, regardless of the shape of the base (such as rectangular, triangular, or polygonal).
Below are the formulas for calculating the volume of a pyramid, with detailed explanations and examples for each type.
Formulas for the Volume of a Pyramid
General Formula Using Base Area and Height:
If the area \( A \) of the base and the height \( h \) of the pyramid are known,
The volume \( V \) can be calculated with the formula:
\( V = \dfrac{1}{3} A \times h \)
In this formula:
- \( A \) is the area of the base of the pyramid
- \( h \) is the height of the pyramid, or the perpendicular distance from the base to the apex
Depending on the shape of the base, the formula for the area \( A \) will vary. Below are specific formulas for pyramids with different base shapes.
Volume of a Rectangular Pyramid
If the base of the pyramid 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 pyramid is then calculated as:
\( V = \dfrac{1}{3} l \times w \times h \)
Example 1: Calculating Volume of a Rectangular Pyramid
Problem: Find the volume of a rectangular pyramid with length \( l = 6 \, \text{cm} \), width \( w = 4 \, \text{cm} \), and height \( h = 10 \, \text{cm} \).
Solution:
- Write down the formula: \( V = \dfrac{1}{3} l \times w \times h \).
- Substitute \( l = 6 \), \( w = 4 \), and \( h = 10 \): \( V = \dfrac{1}{3} \times 6 \times 4 \times 10 \).
- Multiply: \( V = \dfrac{1}{3} \times 240 = 80 \, \text{cm}^3 \).
The volume of the rectangular pyramid is \( 80 \, \text{cm}^3 \).
Volume of a Triangular Pyramid
If the base of the pyramid 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 pyramid is then calculated as:
\( V = \dfrac{1}{3} \times \dfrac{1}{2} b \times h_b \times h \)
Example 2: Calculating Volume of a Triangular Pyramid
Problem: A triangular pyramid has a base length \( b = 5 \, \text{cm} \), base height \( h_b = 6 \, \text{cm} \), and pyramid height \( h = 12 \, \text{cm} \). Find its volume.
Solution:
- Write down the formula: \( V = \dfrac{1}{3} \times \dfrac{1}{2} b \times h_b \times h \).
- Substitute \( b = 5 \), \( h_b = 6 \), and \( h = 12 \): \( V = \dfrac{1}{3} \times \dfrac{1}{2} \times 5 \times 6 \times 12 \).
- Multiply: \( V = \dfrac{1}{3} \times \dfrac{1}{2} \times 360 = 60 \, \text{cm}^3 \).
The volume of the triangular pyramid is \( 60 \, \text{cm}^3 \).
Volume of a Pyramid with a Polygonal Base
For pyramids with polygonal bases (such as pentagonal or hexagonal pyramids), 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:
\( V = \dfrac{1}{3} A \times h \)
Example 3: Calculating Volume of a Hexagonal Pyramid
Problem: A hexagonal pyramid has a base with side length \( s = 4 \, \text{cm} \) 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 pyramid.
Solution:
- Calculate the area of the hexagonal base: \( A = \dfrac{3\sqrt{3}}{2} \times 4^2 \).
- Calculate \( s^2 \): \( A = \dfrac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3} \approx 41.57 \, \text{cm}^2 \) (using \( \sqrt{3} \approx 1.732 \)).
- Now use the volume formula: \( V = \dfrac{1}{3} A \times h \).
- Substitute \( A \approx 41.57 \) and \( h = 15 \): \( V \approx \dfrac{1}{3} \times 41.57 \times 15 \).
- Multiply: \( V \approx 207.85 \, \text{cm}^3 \).
The volume of the hexagonal pyramid is approximately \( 207.85 \, \text{cm}^3 \).
These examples show how to calculate the volume of a pyramid with different base shapes, including rectangular, triangular, and polygonal bases, using the general formula \( V = \dfrac{1}{3} A \times h \).