Calculating the perimeter of a circular segment is a useful skill in geometry. This guide will show you step-by-step how to find the perimeter of a circular segment given the central angle and the radius.
Step 1: Show the Perimeter Formula
The formula for the perimeter \(P\) of a circular segment is given by:
\[ P = \frac{\theta \cdot \pi}{180} \cdot r + 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right) \]
Where:
- \(r\) is the radius of the circle.
- \(\theta\) is the central angle of the segment in degrees.
Step 2: Explain the Formula
In this formula:
- \(\frac{\theta \cdot \pi}{180} \cdot r\) represents the length of the arc of the segment.
- \(2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right)\) represents the straight-line distance across the segment, also known as the chord length.
The perimeter of a circular segment is the sum of the arc length and the chord length.
Step 3: Insert Numbers as an Example
Let's say we have a circular segment with:
- Radius \(r = 10\) units
- Central angle \(\theta = 60^\circ\)
Step 4: Calculate the Final Value
First, we need to find the arc length:
\[ \text{Arc Length} = \frac{\theta \cdot \pi}{180} \cdot r \]
Substitute the values into the formula:
\[ \text{Arc Length} = \frac{60 \cdot \pi}{180} \cdot 10 \]
\[ \text{Arc Length} = \frac{\pi}{3} \cdot 10 \]
\[ \text{Arc Length} = \frac{10\pi}{3} \]
For \(\pi \approx 3.14\):
\[ \text{Arc Length} \approx \frac{10 \cdot 3.14}{3} \]
\[ \text{Arc Length} \approx 10.47 \, \text{units} \]
Next, we need to find the chord length:
\[ \text{Chord Length} = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right) \]
Substitute the values into the formula:
\[ \text{Chord Length} = 2 \cdot 10 \cdot \sin\left(\frac{60}{2}\right) \]
\[ \text{Chord Length} = 20 \cdot \sin(30^\circ) \]
Since \(\sin(30^\circ) = 0.5\):
\[ \text{Chord Length} = 20 \cdot 0.5 \]
\[ \text{Chord Length} = 10 \, \text{units} \]
Finally, we sum the arc length and the chord length to find the perimeter:
\[ P = 10.47 + 10 \]
\[ P = 20.47 \]
So, the perimeter of the circular segment is approximately 20.47 units.
Final Value
The perimeter of a circular segment with a radius of 10 units and a central angle of 60 degrees is approximately 20.47 units.
