Calculating the volume of a spherical sector is a common problem in geometry. A spherical sector is a portion of a sphere defined by two radii and a segment of the sphere. This article will explain the steps to find the volume of a spherical sector using a straightforward formula, including an example calculation.
Volume of a Spherical Sector Formula
To calculate the volume (\( V \)) of a spherical sector, you can use the following formula:
\[ V = \dfrac{2}{3} \cdot \pi \cdot r^2 \cdot h \]
Where:
- \( r \) is the radius of the sphere.
- \( h \) is the height of the spherical sector.
Explanation of the Formula
- The term \( \dfrac{2}{3} \cdot \pi \) is a constant that helps scale the volume of the spherical sector.
- \( r^2 \) represents the square of the radius, which accounts for the area of the circular base of the sector.
- \( h \) represents the height of the spherical sector, which affects the volume based on the distance from the base to the top of the sector.
Step-by-Step Calculation
Let's go through an example to demonstrate how to use this formula.
Example: Calculating the Volume of a Spherical Sector
1. Identify the given values:
- Radius of the sphere (\( r \)) = 4 units
- Height of the spherical sector (\( h \)) = 3 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{2}{3} \cdot \pi \cdot 4^2 \cdot 3 \]
3. Calculate the square of the radius:
\[ 4^2 = 16 \]
4. Substitute the squared value and simplify:
\[ V = \dfrac{2}{3} \cdot \pi \cdot 16 \cdot 3 \]
5. Multiply and simplify further:
\[ V = \dfrac{2}{3} \cdot \pi \cdot 48 \]
\[ V = 32 \cdot \pi \]
6. Calculate the final value (using \(\pi \approx 3.14159\)):
\[ V \approx 32 \cdot 3.14159 \]
\[ V \approx 100.53 \text{ cubic units} \]
By following these steps, you can calculate the volume of any spherical sector given the radius of the sphere and the height of the sector. This formula is essential for solving problems related to spherical shapes in geometry.
