Introduction
Calculating the area of a circular segment can be a bit tricky, but with the right formula, it becomes manageable. This guide will walk you through the process of determining the area of a circular segment using a specific formula. We'll break down each step and provide an example to illustrate the calculations.
The Formula for the Area of a Circular Segment
The area \( A \) of a circular segment can be found using the formula:
\[ A = r^2 \cdot \left( \frac{\theta \cdot \pi}{360^\circ} - \frac{\sin(\theta)}{2} \right) \]
Where:
- \( r \) is the radius of the circle.
- \( \theta \) is the segment angle in degrees.
Explaining the Formula
This formula calculates the area of the segment by combining the area of the sector (a slice of the circle) and subtracting the area of the triangular portion formed by the chord of the segment.
Step-by-Step Calculation
Let's work through an example to illustrate the process.
Example:
Suppose we have a circular segment with a radius \( r = 10 \) units and a segment angle \( \theta = 60^\circ \). We want to find the area of the circular segment.
Step 1: Identify the Given Values
Given:
- Radius \( r = 10 \) units
- Segment angle \( \theta = 60^\circ \)
Step 2: Use the Formula to Find the Area
Substitute the given values into the formula:
\[ A = 10^2 \cdot \left( \frac{60 \cdot \pi}{360^\circ} - \frac{\sin(60^\circ)}{2} \right) \]
Step 3: Perform the Calculation
First, calculate the sector part:
\[ \frac{60 \cdot \pi}{360^\circ} = \frac{\pi}{6} \]
Then, calculate the sine part:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Now, substitute these values back into the formula:
\[ A = 100 \cdot \left( \frac{\pi}{6} - \frac{\frac{\sqrt{3}}{2}}{2} \right) \]
\[ A = 100 \cdot \left( \frac{\pi}{6} - \frac{\sqrt{3}}{4} \right) \]
Convert \( \pi \) to a decimal (approximately \( \pi \approx 3.14159 \)):
\[ \frac{\pi}{6} \approx \frac{3.14159}{6} \approx 0.5236 \]
\[ \frac{\sqrt{3}}{4} \approx \frac{1.73205}{4} \approx 0.433 \]
Now, combine these:
\[ A = 100 \cdot (0.5236 - 0.433) \]
\[ A = 100 \cdot 0.0906 \]
\[ A \approx 9.06 \]
Final Value
For a circular segment with a radius \( r = 10 \) units and a segment angle \( \theta = 60^\circ \), the area is approximately \( 9.06 \) square units.
