To solve these problems, we will use the formulas for the length of an arc and the area of a sector for a circle with a given radius $r$ and an angle $\theta$ in degrees.

Part (a): Length of an Arc

The formula for the length of an arc $L$ that subtends an angle $\theta$ at the center of a circle of radius $r$ is:

$L = 2\pi r \cdot \frac{\theta}{360^\circ}$

Given:

• Radius $r = 12$ cm
• Angle $\theta = 45^\circ$

Substituting these values into the formula:

$L = 2\pi \times 12 \times \frac{45}{360}$

Simplify the expression:

$L = 24\pi \times \frac{1}{8} = 3\pi \, \text{cm}$

So, the length of the arc is $3\pi$ cm.

Part (b): Area of a Sector

The formula for the area $A$ of a sector with angle $\theta$ is:

$A = \pi r^2 \cdot \frac{\theta}{360^\circ}$

Given:

• Radius $r = 12$ cm
• Angle $\theta = 60^\circ$

Substitute these values into the formula:

$A = \pi \times 12^2 \times \frac{60}{360}$

Simplify the expression:

$A = \pi \times 144 \times \frac{1}{6} = 24\pi \, \text{cm}^2$

So, the area of the sector is $24\pi$ cm².

Summary:

• Length of the arc: $3\pi$ cm.
• Area of the sector: $24\pi$ cm².

Would you like more details or have any questions?

Related Questions:

1. How do you convert degrees to radians, and why is it useful in these calculations?
2. What is the formula for the circumference of a circle, and how is it related to arc length?
3. Can you calculate the length of an arc if the radius is different, say 15 cm, and the angle is 90°?
4. What would be the area of a sector with a central angle of 90° in a circle with a radius of 10 cm?
5. How does the area of a sector change if the angle is doubled but the radius remains the same?

Tip: Remember that both the length of an arc and the area of a sector are proportional to the angle subtended at the center of the circle.