Question

written exercises
a 1. a sector of a circle has radius 6 cm and central angle 0.5 radians. find its arc length and area.

1. a sector of a circle has radius 5 cm and central angle 3 radians. find its arc length and area.

Explanation:

Problem 1:

Step1: Recall arc length formula

The formula for the arc length \( s \) of a sector with radius \( r \) and central angle \( \theta \) (in radians) is \( s = r\theta \).
Given \( r = 6 \, \text{cm} \) and \( \theta = 0.5 \, \text{radians} \), we substitute these values into the formula.
\( s = 6\times0.5 \)

Step2: Calculate arc length

\( s = 3 \, \text{cm} \)

Step3: Recall sector area formula

The formula for the area \( A \) of a sector with radius \( r \) and central angle \( \theta \) (in radians) is \( A=\frac{1}{2}r^{2}\theta \).
Substitute \( r = 6 \, \text{cm} \) and \( \theta = 0.5 \, \text{radians} \) into the formula.
\( A=\frac{1}{2}\times6^{2}\times0.5 \)

Step4: Calculate sector area

First, calculate \( 6^{2}=36 \). Then, \( \frac{1}{2}\times36\times0.5 = 18\times0.5=9 \, \text{cm}^2 \)

Step1: Recall arc length formula

The formula for the arc length \( s \) of a sector with radius \( r \) and central angle \( \theta \) (in radians) is \( s = r\theta \).
Given \( r = 5 \, \text{cm} \) and \( \theta = 3 \, \text{radians} \), we substitute these values into the formula.
\( s = 5\times3 \)

Step2: Calculate arc length

\( s = 15 \, \text{cm} \)

Step3: Recall sector area formula

The formula for the area \( A \) of a sector with radius \( r \) and central angle \( \theta \) (in radians) is \( A=\frac{1}{2}r^{2}\theta \).
Substitute \( r = 5 \, \text{cm} \) and \( \theta = 3 \, \text{radians} \) into the formula.
\( A=\frac{1}{2}\times5^{2}\times3 \)

Step4: Calculate sector area

First, calculate \( 5^{2} = 25 \). Then, \( \frac{1}{2}\times25\times3=\frac{75}{2} = 37.5 \, \text{cm}^2 \)

Answer:

Arc length: \( 3 \, \text{cm} \), Area: \( 9 \, \text{cm}^2 \)

Problem 2: