Q4. Find the arc length and the area of the sector for the given problems.

Background

Topic: Arc Length and Area of a Sector (Trigonometry)

This question tests your ability to use the formulas for arc length and area of a sector of a circle, given the central angle and the radius. You will need to work with both degree and radian measures.

Key Terms and Formulas

• Arc Length (): The distance along the curved part of the sector.

• Area of a Sector (): The region enclosed by two radii and the arc.

Formulas:

• Arc Length (when angle is in radians):

• Area of a Sector (when angle is in radians):

• To convert degrees to radians:

Step-by-Step Guidance

(i) For the sector with a central angle of 315° and radius 8 cm:

1. First, convert the central angle from degrees to radians:

2. Calculate the arc length using , where cm and is your answer from step 1.

3. Calculate the area of the sector using .

(ii) For the sector with a central angle of radians and radius 13 ft:

1. Since the angle is already in radians (), you can use it directly in the formulas.

2. Calculate the arc length using , where ft.

3. Calculate the area of the sector using .

Try solving on your own before revealing the answer!