Calculating the surface area of a hemisphere, which is half of a sphere, is a fundamental geometric concept. This guide will walk you through the process step-by-step using the specific formula for the surface area of a hemisphere.
Understanding the Surface Area Formula
The surface area (SA) of a hemisphere can be calculated using the following formula:
\[ SA = 3 \cdot \pi \cdot r^2 \]
Where:
- \( r \) is the radius of the hemisphere.
- \( \pi \) (pi) is a mathematical constant approximately equal to 3.14159.
Explaining the Formula
- The term \( 3 \cdot \pi \cdot r^2 \) represents the total surface area of the hemisphere.
- \( 3 \) is a constant that relates to the geometry of the hemisphere.
- \( \pi \) is a constant that appears in formulas involving circles, spheres, and hemispheres.
- \( r^2 \) indicates that the radius is squared, meaning it is multiplied by itself.
The formula accounts for both the curved surface area of the hemisphere and the flat circular base.
Step-by-Step Calculation
Let's calculate the surface area of a hemisphere with a given radius.
Example: Calculating the Surface Area of a Hemisphere with a Radius of 4 Units
1. Identify the given value:
Radius (\( r \)) = 4 units
2. Substitute the given value into the formula:
\[ SA = 3 \cdot \pi \cdot 4^2 \]
3. Calculate the radius squared:
\[ 4^2 = 16 \]
4. Substitute and simplify:
\[ SA = 3 \cdot \pi \cdot 16 \]
5. Multiply by \( \pi \):
\[ SA = 3 \cdot 3.14159 \cdot 16 \]
6. Simplify the multiplication:
\[ SA \approx 3 \cdot 3.14159 \cdot 16 = 9.42477 \cdot 16 = 150.796 \]
Final Value
The surface area of a hemisphere with a radius of 4 units is approximately 150.80 square units.
