To solve for the area of a sector and the length of the arc of a circle, where the radius $r$ is 21 cm and the central angle $\theta$ is $60^\circ$, we use the following formulas:

1. Area of the Sector

The formula for the area $A$ of a sector is: $A = \frac{\theta}{360^\circ} \times \pi r^2$

Substituting the values:

$A = \frac{60^\circ}{360^\circ} \times \pi \times (21)^2$ $A = \frac{1}{6} \times \pi \times 441$ $A = 73.5 \pi \approx 230.91 \text{ cm}^2$

So, the area of the sector is approximately $230.91 \, \text{cm}^2$.

2. Length of the Arc

The formula for the length $L$ of the arc is: $L = \frac{\theta}{360^\circ} \times 2 \pi r$

Substituting the values:

$L = \frac{60^\circ}{360^\circ} \times 2 \pi \times 21$ $L = \frac{1}{6} \times 42 \pi$ $L = 7 \pi \approx 21.99 \, \text{cm}$

Thus, the length of the arc is approximately $21.99 \, \text{cm}$.

Summary

• Area of the Sector: $230.91 \, \text{cm}^2$
• Length of the Arc: $21.99 \, \text{cm}$

Would you like further details, or do you have any questions?

Related Questions:

1. What is the area of a sector if the radius is 15 cm and the angle is 45°?
2. How does changing the central angle affect the area of a sector?
3. What is the length of an arc when the radius is 10 cm and the central angle is 90°?
4. If the radius of a circle doubles, how does the area of a sector change?
5. What is the formula to find the area of a segment, and how does it differ from a sector?

Tip:

Remember that the area and arc length of a sector are proportional to the central angle, so increasing the angle will increase both the area and the arc length proportionally.