An open spherical sector is a part of a sphere bounded by a spherical cap and a curved surface extending from a circle on the sphere. To find the surface area of such a shape, you can use a specific formula that takes into account its unique geometry. This article will guide you through the process, including step-by-step calculations and an example.
Formula to Calculate the Surface Area of an Open Spherical Sector
The surface area (\( SA \)) of an open spherical sector can be determined using the following formula:
\[ SA = 2 \cdot \pi \cdot r \cdot h + \pi \cdot r_1 \cdot r + \pi \cdot r_2 \cdot r \]
Where:
- \( SA \) is the surface area of the open 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 (upper base).
- \( r_2 \) is the radius of the sector's base (lower base).
Explanation of the Formula
The formula for the surface area of an open 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. \( \pi \cdot r_1 \cdot r \): This part calculates the area of the spherical cap (upper base).
3. \( \pi \cdot r_2 \cdot r \): This part calculates the area of the base of the spherical sector.
Example Calculation
Let’s go through an example to illustrate how to use this formula.
Given:
- \( r = 10 \) units (the radius of the sphere)
- \( h = 6 \) units (the height of the spherical sector)
- \( r_1 = 8 \) units (the radius of the spherical cap)
- \( r_2 = 6 \) units (the radius of the sector's base)
We want to find the surface area of the open spherical sector.
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- \( r = 10 \) units
- \( h = 6 \) units
- \( r_1 = 8 \) units
- \( r_2 = 6 \) units
Step 2: Use the Surface Area Formula
\[ SA = 2 \cdot \pi \cdot r \cdot h + \pi \cdot r_1 \cdot r + \pi \cdot r_2 \cdot r \]
Step 3: Substitute the Given Values into the Formula
\[ SA = 2 \cdot \pi \cdot 10 \cdot 6 + \pi \cdot 8 \cdot 10 + \pi \cdot 6 \cdot 10 \]
Step 4: Calculate the Lateral Surface Area
\[ 2 \cdot \pi \cdot 10 \cdot 6 = 120 \cdot \pi \]
Step 5: Calculate the Area of the Spherical Cap (Upper Base)
\[ \pi \cdot 8 \cdot 10 = 80 \cdot \pi \]
Step 6: Calculate the Area of the Sector’s Base (Lower Base)
\[ \pi \cdot 6 \cdot 10 = 60 \cdot \pi \]
Step 7: Sum the Three Parts to Find the Total Surface Area
\[ SA = 120 \cdot \pi + 80 \cdot \pi + 60 \cdot \pi \]
\[ SA = \pi \cdot (120 + 80 + 60) \]
\[ SA = \pi \cdot 260 \]
Step 8: Calculate the Final Value
\[ SA \approx 3.14159 \cdot 260 \approx 816.81 \]
Final Value
The surface area of an open spherical sector with a sphere radius of 10 units, height of 6 units, spherical cap radius of 8 units, and sector base radius of 6 units is approximately \( 816.81 \) square units.
