Calculating the volume of a spherical cap is a common problem in geometry. A spherical cap is a portion of a sphere cut off by a plane. This article will explain the steps to find the volume of a spherical cap using a straightforward formula, including an example calculation.
Volume of a Spherical Cap Formula
To calculate the volume (\( V \)) of a spherical cap, you can use the following formula:
\[ V = \dfrac{1}{3} \cdot \pi \cdot r^3 \cdot \left( 2 - 3 \cdot \sin(\theta) + \sin^3(\theta) \right)\]
Where:
- \( r \) is the radius of the sphere.
- \( \theta \) is the angle (in radians) that the cap subtends at the center of the sphere.
Explanation of the Formula
- The term \( \dfrac{1}{3} \) is a constant that helps scale the volume of the spherical cap.
- \( \pi \) 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 sphere.
- The expression \( 2 - 3 \cdot \sin(\theta) + \sin^3(\theta) \) adjusts the volume based on the angle subtended by the cap.
Step-by-Step Calculation
Let's go through an example to demonstrate how to use this formula.
Example: Calculating the Volume of a Spherical Cap
1. Identify the given values:
- Radius of the sphere (\( r \)) = 5 units
- Angle subtended by the cap (\( \theta \)) = \( \dfrac{\pi}{6} \) radians (30 degrees)
2. Substitute the values into the volume formula:
\[ V = \dfrac{1}{3} \cdot \pi \cdot 5^3 \cdot \left( 2 - 3 \cdot \sin\left( \dfrac{\pi}{6} \right) + \sin^3\left( \dfrac{\pi}{6} \right) \right)\]
3. Calculate the cube of the radius:
\[ 5^3 = 125 \]
4. Substitute the value and simplify:
\[ V = \dfrac{1}{3} \cdot \pi \cdot 125 \cdot \left( 2 - 3 \cdot \sin\left( \dfrac{\pi}{6} \right) + \sin^3\left( \dfrac{\pi}{6} \right) \right)\]
5. Evaluate the trigonometric functions:
\[ \sin\left( \dfrac{\pi}{6} \right) = \dfrac{1}{2} \]
\[ \sin^3\left( \dfrac{\pi}{6} \right) = \left( \dfrac{1}{2} \right)^3 = \dfrac{1}{8}\]
6. Substitute these values back into the expression:
\[ V = \dfrac{1}{3} \cdot \pi \cdot 125 \cdot \left( 2 - 3 \cdot \dfrac{1}{2} + \dfrac{1}{8} \right)\]
7. Simplify the expression inside the parentheses:
\[ 2 - \dfrac{3}{2} + \dfrac{1}{8} = \dfrac{16}{8} - \dfrac{12}{8} + \dfrac{1}{8} = \dfrac{5}{8}\]
8. Substitute and simplify:
\[ V = \dfrac{1}{3} \cdot \pi \cdot 125 \cdot \dfrac{5}{8}\]
9. Multiply the terms:
\[ V = \dfrac{1}{3} \cdot \pi \cdot \dfrac{625}{8}\]
\[ V = \dfrac{625 \cdot \pi}{24}\]
10. Calculate the final value using \( \pi \approx 3.14159 \):
\[ V \approx \dfrac{625 \cdot 3.14159}{24} \]
\[ V \approx 81.68 \text{ cubic units}\]
Final Volume
The volume of a spherical cap with a radius of 5 units and an angle of \( \dfrac{\pi}{6} \) radians is approximately 81.68 cubic units.
