Finding the surface area of a closed spherical sector involves a specific formula that accounts for the geometry of this unique shape. This article will guide you through the process, providing step-by-step instructions and a practical example.
Formula to Calculate the Surface Area of a Closed Spherical Sector
The surface area (\( SA \)) of a closed spherical sector can be determined using the following formula:
\[ SA = 2 \cdot \pi \cdot r \cdot h + \pi \cdot r_1^2 + \pi \cdot r_2 \cdot r \]
Where:
- \( SA \) is the surface area of the closed spherical sector.
- \( r \) is the radius of the sphere.
- \( h \) is the height of the spherical sector from the center of the sphere to the base.
- \( r_1 \) is the radius of the spherical cap (circular base).
- \( r_2 \) is the radius of the sector's base.
Explanation of the Formula
The formula for the surface area of a closed spherical sector consists of three parts:
1. \( 2 \cdot \pi \cdot r \cdot h \): This part calculates the lateral surface area of the spherical sector.
2. \( \pi \cdot r_1^2 \): This part calculates the area of the spherical cap (circular base).
3. \( \pi \cdot r_2 \cdot r \): This part calculates the area of the spherical segment (surface area of the curved part connected to the base).
Example Calculation
Let's go through an example to illustrate how to use this formula.
Given:
- \( r = 8 \) units (the radius of the sphere)
- \( h = 4 \) units (the height of the spherical sector)
- \( r_1 = 6 \) units (the radius of the spherical cap)
- \( r_2 = 5 \) units (the radius of the sector's base)
We want to find the surface area of the closed spherical sector.
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- \( r = 8 \) units
- \( h = 4 \) units
- \( r_1 = 6 \) units
- \( r_2 = 5 \) units
Step 2: Use the Surface Area Formula
\[ SA = 2 \cdot \pi \cdot r \cdot h + \pi \cdot r_1^2 + \pi \cdot r_2 \cdot r \]
Step 3: Substitute the Given Values into the Formula
\[ SA = 2 \cdot \pi \cdot 8 \cdot 4 + \pi \cdot 6^2 + \pi \cdot 5 \cdot 8 \]
Step 4: Calculate the Lateral Surface Area
\[ 2 \cdot \pi \cdot 8 \cdot 4 = 64 \cdot \pi \]
Step 5: Calculate the Area of the Spherical Cap
\[ \pi \cdot 6^2 = \pi \cdot 36 \]
Step 6: Calculate the Area of the Spherical Segment
\[ \pi \cdot 5 \cdot 8 = 40 \cdot \pi \]
Step 7: Sum the Three Parts to Find the Total Surface Area
\[ SA = 64 \cdot \pi + \pi \cdot 36 + 40 \cdot \pi \]
\[ SA = \pi \cdot (64 + 36 + 40) \]
\[ SA = \pi \cdot 140 \]
Step 8: Calculate the Final Value
\[ SA \approx 3.14159 \cdot 140 \approx 439.82 \]
Final Value
The surface area of a closed spherical sector with a sphere radius of 8 units, height of 4 units, spherical cap radius of 6 units, and sector base radius of 5 units is approximately \( 439.82 \) square units.
