Finding the surface area of a spherical zone involves using a specific formula tailored to the geometric properties of the zone. This article will guide you through calculating the surface area of a spherical zone with clear steps and a practical example.
Formula to Calculate the Surface Area of a Spherical Zone
The surface area (\(SA\)) of a spherical zone can be determined using the following formula:
\[ SA = 2 \cdot \pi \cdot r \cdot h + \pi \cdot r_1^2 + \pi \cdot r_2^2 \]
Where:
- \( SA \) is the surface area of the spherical zone.
- \( r \) is the radius of the sphere.
- \( h \) is the height of the spherical zone.
- \( r_1 \) is the radius of the upper base of the spherical zone.
- \( r_2 \) is the radius of the lower base of the spherical zone.
Explanation of the Formula
The formula for the surface area of a spherical zone consists of three parts:
1. \( 2 \cdot \pi \cdot r \cdot h \): This part calculates the lateral surface area of the spherical zone.
2. \( \pi \cdot r_1^2 \): This part calculates the area of the upper base of the spherical zone.
3. \( \pi \cdot r_2^2 \): This part calculates the area of the lower base of the spherical zone.
Example Calculation
Let's go through an example to illustrate how to use this formula.
Given:
- \( r = 10 \) units (the radius of the sphere)
- \( h = 5 \) units (the height of the spherical zone)
- \( r_1 = 6 \) units (the radius of the upper base)
- \( r_2 = 8 \) units (the radius of the lower base)
We want to find the surface area of the spherical zone.
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- \( r = 10 \) units
- \( h = 5 \) units
- \( r_1 = 6 \) units
- \( r_2 = 8 \) units
Step 2: Use the Surface Area Formula
\[ SA = 2 \cdot \pi \cdot r \cdot h + \pi \cdot r_1^2 + \pi \cdot r_2^2 \]
Step 3: Substitute the Given Values into the Formula
\[ SA = 2 \cdot \pi \cdot 10 \cdot 5 + \pi \cdot 6^2 + \pi \cdot 8^2 \]
Step 4: Calculate the Lateral Surface Area
\[ 2 \cdot \pi \cdot 10 \cdot 5 = 100 \cdot \pi \]
Step 5: Calculate the Area of the Upper Base
\[ \pi \cdot 6^2 = \pi \cdot 36 \]
Step 6: Calculate the Area of the Lower Base
\[ \pi \cdot 8^2 = \pi \cdot 64 \]
Step 7: Sum the Three Parts to Find the Total Surface Area
\[ SA = 100 \cdot \pi + \pi \cdot 36 + \pi \cdot 64 \]
\[ SA = \pi \cdot (100 + 36 + 64) \]
\[ SA = \pi \cdot 200 \]
Step 8: Calculate the Final Value
\[ SA \approx 3.14159 \cdot 200 \approx 628.32 \]
Final Value
The surface area of a spherical zone with a sphere radius of 10 units, height of 5 units, upper base radius of 6 units, and lower base radius of 8 units is approximately \( 628.32 \) square units.
