Calculating the perimeter of a circular sector is a crucial concept in geometry. This guide will show you step-by-step how to find the perimeter of a circular sector given the perimeter of the full circle and the central angle.
Step 1: Show the Perimeter Formula
The formula for the perimeter \(P\) of a circular sector is given by:
\[ P = \frac{\theta}{360^\circ} \cdot \pi \cdot 2 \cdot r + 2 \cdot r = \frac{\theta}{360^\circ} \cdot \pi \cdot d + d \]
Where:
- \(r\) is the radius of the circle.
- \(\theta\) is the central angle of the sector in degrees.
- \(d\) is the diameter of the circle, which is \(2r\).
Step 2: Explain the Formula
In this formula:
- \(\frac{\theta}{360^\circ} \cdot \pi \cdot 2 \cdot r\) represents the length of the arc of the sector.
- \(2 \cdot r\) represents the sum of the two radii that form the boundaries of the sector.
The perimeter of a circular sector is the sum of the arc length and the two radii.
Step 3: Insert Numbers as an Example
Let's say we have a circular sector with:
- The perimeter of the full circle \(C = 50\) units
- Central angle \(\theta = 90^\circ\)
Step 4: Calculate the Final Value
First, we need to find the radius \(r\) of the circle using the perimeter \(C\):
\[ C = 2 \cdot \pi \cdot r \]
\[ 50 = 2 \cdot \pi \cdot r \]
\[ r = \frac{50}{2\pi} \]
\[ r = \frac{50}{2 \cdot 3.14} \]
\[ r \approx 7.96 \, \text{units} \]
Next, we need to find the arc length:
\[ \text{Arc Length} = \frac{\theta}{360^\circ} \cdot \pi \cdot 2 \cdot r \]
Substitute the values into the formula:
\[ \text{Arc Length} = \frac{90^\circ}{360^\circ} \cdot \pi \cdot 2 \cdot 7.96 \]
\[ \text{Arc Length} = \frac{1}{4} \cdot \pi \cdot 15.92 \]
\[ \text{Arc Length} = \frac{15.92\pi}{4} \]
\[ \text{Arc Length} = 3.98\pi \]
Now, we add the two radii:
\[ P = 3.98\pi + 2 \cdot 7.96 \]
\[ P = 3.98 \cdot 3.14 + 15.92 \]
\[ P \approx 12.49 + 15.92 \]
\[ P \approx 28.41 \]
So, the perimeter of the circular sector is approximately 28.41 units.
Final Value
The perimeter of a circular sector with a perimeter of the full circle of 50 units and a central angle of 90 degrees is approximately 28.41 units.
