IDNLearn.com makes it easy to get reliable answers from experts and enthusiasts alike. Our platform offers comprehensive and accurate responses to help you make informed decisions on any topic.

Find the perimeter of the sector for [tex]\frac{1}{8}[/tex] of a circle in terms of [tex]\pi[/tex].

Sagot :

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.