Calculating the volume of a spherical zone is a common problem in geometry. A spherical zone, also known as a spherical segment, is a portion of a sphere bounded by two parallel planes. This article will explain the steps to find the volume of a spherical zone using a straightforward formula, including an example calculation.
Volume of a Spherical Zone Formula
To calculate the volume (\( V \)) of a spherical zone, you can use the following formula:
\[ V = \dfrac{1}{6} \cdot \pi \cdot h \cdot (3 \cdot r_1^2 + 3 \cdot r_2^2 + h^2)\]
Where:
- \( h \) is the height of the spherical zone.
- \( r_1 \) is the radius of the top base of the spherical zone.
- \( r_2 \) is the radius of the bottom base of the spherical zone.
Explanation of the Formula
- The term \( \dfrac{1}{6} \cdot \pi \) is a constant that helps scale the volume of the spherical zone.
- \( h \) represents the height of the spherical zone, which affects the volume based on the vertical distance between the two parallel planes.
- \( 3 \cdot r_1^2 + 3 \cdot r_2^2 + h^2 \) accounts for the radii of the bases and the height, which together determine the overall volume of the zone.
Step-by-Step Calculation
Let's go through an example to demonstrate how to use this formula.
Example: Calculating the Volume of a Spherical Zone
1. Identify the given values:
- Height of the spherical zone (\( h \)) = 5 units
- Radius of the top base (\( r_1 \)) = 2 units
- Radius of the bottom base (\( r_2 \)) = 3 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{1}{6} \cdot \pi \cdot 5 \cdot (3 \cdot 2^2 + 3 \cdot 3^2 + 5^2)\]
3. Calculate the squares of the radii and height:
\[ 2^2 = 4 \]
\[ 3^2 = 9 \]
\[ 5^2 = 25 \]
4. Substitute the squared values and simplify:
\[ V = \dfrac{1}{6} \cdot \pi \cdot 5 \cdot (3 \cdot 4 + 3 \cdot 9 + 25)\]
\[ V = \dfrac{1}{6} \cdot \pi \cdot 5 \cdot (12 + 27 + 25)\]
\[ V = \dfrac{1}{6} \cdot \pi \cdot 5 \cdot 64\]
5. Multiply and simplify further:
\[ V = \dfrac{1}{6} \cdot \pi \cdot 320\]
\[ V = \dfrac{320 \pi}{6}\]
\[ V = \dfrac{160 \pi}{3}\]
6. Calculate the final value (using \(\pi \approx 3.14159\)):
\[ V \approx \dfrac{160 \cdot 3.14159}{3}\]
\[ V \approx \dfrac{502.654}{3}\]
\[ V \approx 167.55 \text{ cubic units}\]
By following these steps, you can calculate the volume of any spherical zone given the height of the zone and the radii of the top and bottom bases.
