A regular hexahedron, more commonly known as a cube, is a three-dimensional shape with six equal square faces. Calculating the volume of a cube is a straightforward task, essential in various fields such as architecture, engineering, and everyday measurements.
Volume Formula for a Regular Hexahedron
The volume \( V \) of a regular hexahedron (cube) is determined by the formula:
\[ V = a^3 \]
Where:
- \( V \) is the volume.
- \( a \) is the length of an edge of the cube.
This formula calculates the volume by cubing the length of one of the cube’s edges.
Step-by-Step Calculation
Let's walk through the calculation with an example to make the process clear.
Given:
- Edge length \( a = 4 \) units
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- \( a = 4 \) units
Step 2: Substitute the Value into the Volume Formula
Using the formula:
\[ V = a^3 \]
Substitute \( a = 4 \):
\[ V = 4^3 \]
Step 3: Calculate the Cube of the Edge Length
Calculate \( 4^3 \):
\[ 4 \times 4 \times 4 = 64 \]
Final Value
The volume of a regular hexahedron (cube) with an edge length of 4 units is \( 64 \) cubic units.
Detailed Example Calculation
Let’s go through the calculation with detailed steps:
1. Substitute the Edge Length into the Formula:
\[ V = 4^3 \]
2. Calculate the Cube of the Edge Length:
\[ 4 \times 4 \times 4 = 64 \]
Conclusion
Finding the volume of a regular hexahedron is a simple yet crucial calculation. By cubing the length of any of the cube’s edges, you can quickly determine its volume. This method is invaluable in practical scenarios where understanding the space occupied by a cube is necessary.
Additional Example
To illustrate further, let’s consider another example:
Example 2:
- Edge length \( a = 5 \) units
Calculation:
1. Substitute into the formula:
\[ V = 5^3 \]
2. Calculate:
\[ 5 \times 5 \times 5 = 125 \]
Thus, the volume of a cube with an edge length of 5 units is \( 125 \) cubic units.
This straightforward formula \( V = a^3 \) provides a quick and efficient way to calculate the volume of any regular hexahedron, making it a fundamental tool in mathematics and various practical applications.
