To find the perimeter of a sector that represents [tex]\(\frac{1}{8}\)[/tex] of a circle, follow these steps:
1. Understand the parameters:
- The circle is divided into 8 equal parts.
- We'll denote the angle of the sector as [tex]\(\theta\)[/tex].
- Since [tex]\(\frac{1}{8}\)[/tex] of a full circle corresponds to an angle:
[tex]\[
\theta = \frac{1}{8} \times 2\pi = \frac{\pi}{4} \text{ radians}
\][/tex]
- Let [tex]\( r \)[/tex] be the radius of the circle.
2. Calculate the length of the arc:
- The length of the arc of a sector is given by:
[tex]\[
\text{Arc length} = r \times \theta
\][/tex]
- Substituting [tex]\(\theta = \frac{\pi}{4}\)[/tex], we get:
[tex]\[
\text{Arc length} = r \times \frac{\pi}{4}
\][/tex]
- If we assume [tex]\( r = 1 \)[/tex] as a symbolic representation, the arc length will be:
[tex]\[
\text{Arc length} = \frac{\pi}{4}
\][/tex]
- Numerically, this central angle approximation is:
[tex]\[
\text{Arc length} \approx 0.7853981633974483
\][/tex]
3. Calculate the perimeter of the sector:
- The perimeter of the sector includes the arc length and the two radii that form the sector:
[tex]\[
\text{Perimeter} = \text{Arc length} + 2r
\][/tex]
- Substituting the values:
[tex]\[
\text{Perimeter} = \frac{\pi}{4} + 2 \times 1
\][/tex]
- Which simplifies to:
[tex]\[
\text{Perimeter} = \frac{\pi}{4} + 2
\][/tex]
- Numerically, this evaluates to:
[tex]\[
\text{Perimeter} \approx 2.7853981633974483
\][/tex]
Therefore, the perimeter of the sector for [tex]\(\frac{1}{8}\)[/tex] of a circle, assuming the radius [tex]\( r = 1 \)[/tex], is [tex]\(\frac{\pi}{4} + 2\)[/tex], or approximately 2.7853981633974483.
