Calculating the perimeter of a circular sector is an essential concept in geometry. This guide will show you step-by-step how to find the perimeter of a circular sector given the radius 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:
- Radius \(r = 5\) units
- Central angle \(\theta = 90^\circ\)
Step 4: Calculate the Final Value
First, 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:
\[ \text{Arc Length} = \frac{90^\circ}{360^\circ} \cdot \pi \cdot 2 \cdot 5 \]
\[ \text{Arc Length} = \frac{1}{4} \cdot \pi \cdot 10 \]
\[ \text{Arc Length} = \frac{10\pi}{4} \]
\[ \text{Arc Length} = 2.5\pi \]
Now, we add the two radii:
\[ P = 2.5\pi + 2 \cdot 5 \]
\[ P = 2.5\pi + 10 \]
For \(\pi \approx 3.14\):
\[ P \approx 2.5 \cdot 3.14 + 10 \]
\[ P \approx 7.85 + 10 \]
\[ P \approx 17.85 \]
So, the perimeter of the circular sector is approximately 17.85 units.
Final Value
The perimeter of a circular sector with a radius of 5 units and a central angle of 90 degrees is approximately 17.85 units.
