A circle has radius 12 cm. Find, in exact form:
a the length of an arc that subtends an angle at the centre of 45°,
b the area of a sector in which the angle at the centre is 60°.

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.

A circle has radius 12 cm. Find, in exact form: a. the length of an arc that subtends an angle at the centre of 45°, b. the area of a sector in which the angle at the centre is 60°.

Learn how to calculate the exact arc length and sector area of a circle with radius 12 cm. Solve for the arc length with an angle of 45° and the sector area with an angle of 60°. Suitable for students in Grades 8-10.

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