Calculating the volume of a hemisphere, which is half of a sphere, is a common problem in geometry. This article will explain the steps to find the volume of a hemisphere using a straightforward formula, including an example calculation.
Volume of a Hemisphere Formula
To calculate the volume (\( V \)) of a hemisphere, you can use the following formula:
\[ V = \dfrac{2}{3} \cdot \pi \cdot r^3 \]
Where:
- \( r \) is the radius of the hemisphere.
Explanation of the Formula
- The term \( \dfrac{2}{3} \) is a constant that reflects the fact that a hemisphere is half of a sphere.
- \( \pi \) is a mathematical constant approximately equal to 3.14159.
- \( r^3 \) represents the cube of the radius, which scales the volume based on the size of the hemisphere.
Step-by-Step Calculation
Let's go through an example to demonstrate how to use this formula.
Example: Calculating the Volume of a Hemisphere
1. Identify the given value:
- Radius of the hemisphere (\( r \)) = 4 units
2. Substitute the value into the volume formula:
\[ V = \dfrac{2}{3} \cdot \pi \cdot 4^3 \]
3. Calculate the cube of the radius:
\[ 4^3 = 64 \]
4. Substitute and simplify:
\[ V = \dfrac{2}{3} \cdot \pi \cdot 64 \]
5. Multiply the terms:
\[ V = \dfrac{128}{3} \cdot \pi \]
6. Calculate the final value using \( \pi \approx 3.14159 \):
\[ V \approx \dfrac{128}{3} \cdot 3.14159 \]
\[ V \approx 134.041 \text{ cubic units} \]
Final Volume
The volume of a hemisphere with a radius of 4 units is approximately 134.041 cubic units.
