Introduction
Calculating the area of a hexagon, a six-sided polygon, can be simplified when you know the side length and the area of the inscribed circle (the circle that touches all sides of the polygon at their midpoints). This guide will take you through the process step-by-step, making it easy to understand and apply.
Understanding the Hexagon
A hexagon is a six-sided polygon, and when all its sides are equal, it is known as a regular hexagon. The area of a regular hexagon can be determined using the length of its sides and the radius of its inscribed circle.
Formulas Needed
1. Radius of the Inscribed Circle:
\[ \text{Area of Circle} = \pi r^2 \]
\[ r = \sqrt{\frac{\text{Area of Circle}}{\pi}} \]
2. Area of the Hexagon:
\[ \text{Area of Hexagon} = \frac{n}{2} \times L \times r \]
Where \( L \) is the length of one side of the hexagon.
Step-by-Step Calculation
Let's walk through the calculation process with an example where the side length \( L \) is 6 units, and the area of the inscribed circle is 54 square units.
Step 1: Identify the Given Values
Given:
- Side length \( L \)
- Area of the inscribed circle
Step 2: Calculate the Radius of the Inscribed Circle
Using the area of the circle formula:
\[ \text{Area of Circle} = \pi r^2 \]
\[ 54 = \pi r^2 \]
Solving for \( r \):
\[ r^2 = \frac{54}{\pi} \]
\[ r^2 = \frac{54}{3.14159} \]
\[ r^2 \approx 17.2 \]
\[ r \approx \sqrt{17.2} \]
\[ r \approx 4.15 \text{ units} \]
Step 3: Use the Side Length to Calculate the Area of the Hexagon
Using the area of a hexagon formula:
\[ \text{Area of Hexagon} = \frac{n}{2} L \times r \]
\[ \text{Area of Hexagon} = \frac{6}{2} 6 \times 4.15 \]
\[ \text{Area} \approx 74.7 \text{ square units} \]
Final Value
The area of the hexagon with each side measuring 6 units and an inscribed circle area of 54 square units is approximately 74.7 square units.
