Introduction
Determining the radius of a circle from the area of a circular segment and its angle can be a bit complex, but it's manageable with algebra. This guide will walk you through the process of isolating the radius from the formula for the area of a circular segment. 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 is given by:
\[ A = r^2 \cdot \left( \frac{\theta \cdot \pi}{360^\circ} - \frac{\sin(\theta)}{2} \right) \]
Where:
- \( r \) is the radius of the circle.
- \( \theta \) is the segment angle in degrees.
Isolating the Radius \( r \)
To determine the radius \( r \), we need to isolate \( r \) in the formula. Start by rearranging the formula to solve for \( r \):
\[ A = r^2 \cdot \left( \frac{\theta \cdot \pi}{360^\circ} - \frac{\sin(\theta)}{2} \right) \]
Divide both sides by \( \left( \frac{\theta \cdot \pi}{360^\circ} - \frac{\sin(\theta)}{2} \right) \):
\[ r^2 = \frac{A}{\left( \frac{\theta \cdot \pi}{360^\circ} - \frac{\sin(\theta)}{2} \right)} \]
Take the square root of both sides to solve for \( r \):
\[ r = \sqrt{\frac{A}{\left( \frac{\theta \cdot \pi}{360^\circ} - \frac{\sin(\theta)}{2} \right)}} \]
Step-by-Step Calculation
Let's work through an example to illustrate the process.
Example:
Suppose we have a circular segment with an area \( A = 50 \) square units and a segment angle \( \theta = 90^\circ \). We want to find the radius of the circle.
Step 1: Identify the Given Values
Given:
- Area \( A = 50 \) square units
- Segment angle \( \theta = 90^\circ \)
Step 2: Substitute the Given Values into the Formula
First, calculate the terms inside the parenthesis:
\[ \frac{\theta \cdot \pi}{360^\circ} = \frac{90 \cdot \pi}{360^\circ} = \frac{\pi}{4} \]
\[ \sin(90^\circ) = 1 \]
So the formula becomes:
\[ r = \sqrt{\frac{50}{\left( \frac{\pi}{4} - \frac{1}{2} \right)}} \]
Step 3: Simplify the Denominator
Convert \( \pi \) to a decimal (approximately \( \pi \approx 3.14159 \)):
\[ \frac{\pi}{4} \approx \frac{3.14159}{4} \approx 0.7854 \]
\[ \frac{1}{2} = 0.5 \]
So the denominator is:
\[ 0.7854 - 0.5 = 0.2854 \]
Step 4: Calculate the Radius
Now, substitute back into the formula:
\[ r = \sqrt{\frac{50}{0.2854}} \]
\[ r = \sqrt{175.18} \]
\[ r \approx 13.24 \]
Final Value
For a circular segment with an area of 50 square units and a segment angle of 90 degrees, the radius of the circle is approximately 13.24 units.
