Arc Length and Area of a Sector – Step-by-Step Guidance

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 ($s$): The distance along the curved part of the sector.

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

Formulas:

• Arc Length (when angle is in radians): $s = r\theta$

• Area of a Sector (when angle is in radians): $A = \frac{1}{2} r^2 \theta$

• To convert degrees to radians: $\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}$

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: $\theta = 315^\circ \times \frac{\pi}{180}$

2. Calculate the arc length using $s = r\theta$, where $r = 8$ cm and $\theta$ is your answer from step 1.

3. Calculate the area of the sector using $A = \frac{1}{2} r^2 \theta$.

(ii) For the sector with a central angle of $\frac{3\pi}{4}$ radians and radius 13 ft:

1. Since the angle is already in radians ($\theta = \frac{3\pi}{4}$), you can use it directly in the formulas.

2. Calculate the arc length using $s = r\theta$, where $r = 13$ ft.

3. Calculate the area of the sector using $A = \frac{1}{2} r^2 \theta$.

Try solving on your own before revealing the answer!

Final Answers:

(i) For the 315° sector:

Arc length: $s \approx 44$ cm Area: $A \approx 100.5$ cm2

(ii) For the $\frac{3\pi}{4}$ sector:

Arc length: $s \approx 30.7$ ft Area: $A \approx 133.1$ ft2

We used the formulas for arc length and area, making sure to convert degrees to radians where necessary.