This step-by-step guide will walk you through the process using real whole numbers, ensuring it's easy to follow and understand.
Step 1: Understand the Key Formula
The formula to calculate the area (\(A\)) of a regular polygon with \(n\) sides of length \(s\) and an inner radius (also called the apothem) \(r\) is:
\[ A = \frac{1}{2} \times P \times r \]
where \(P\) is the perimeter of the polygon.
Step 2: Calculate the Perimeter
First, we need to calculate the perimeter (\(P\)) of the pentagon. The perimeter is the total length of all the sides. For a regular pentagon with side length \(s\):
\[ P = n \times s \]
Since a pentagon has 5 sides (\(n = 5\)):
\[ P = 5 \times s \]
Step 3: Insert Actual Numbers
Let's assume the side length (\(s\)) is 8 units and the inner radius (\(r\)) is 5 units.
Step 4: Calculate the Perimeter with Given Values
Substitute the given side length into the perimeter formula:
\[ P = 5 \times 8 \]
\[ P = 40 \, \text{units} \]
Step 5: Calculate the Area
Now, use the formula to calculate the area with the given inner radius (\(r\)) and the calculated perimeter (\(P\)):
\[ A = \frac{1}{2} \times P \times r \]
Substitute the known values:
\[ A = \frac{1}{2} \times 40 \times 5 \]
\[ A = \frac{1}{2} \times 200 \]
Perform the multiplication:
\[ A = 100 \, \text{square units} \]
Summary of Steps
1. **Understand the formula**: \(A = \frac{1}{2} \times P \times r\)
2. **Calculate the perimeter**: \(P = n \times s\)
3. **Use real whole numbers**: For example, \(s = 8\) units, \(r = 5\) units
4. **Calculate the perimeter**: \(P = 40\) units
5. **Calculate the area**: \(A = 100\) square units
By following these steps, you can easily determine the area of a pentagon given the side length and the inner radius. Understanding these geometric principles will enhance your problem-solving skills and mathematical knowledge. Happy learning!
