To solve this worksheet, we need to calculate the
area of a sector of a circle. The formula for the area of a sector is:
$$
\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2
$$
Where:
- $\theta$ is the central angle in degrees,
- $r$ is the radius of the circle.
We will apply this formula to each of the six sectors.
---
🔹 Problem 1:
- Angle: $35^\circ$
- Radius: $10$
$$
\text{Area} = \frac{35}{360} \times \pi \times 10^2 = \frac{35}{360} \times \pi \times 100
= \frac{3500}{360} \pi \approx 9.722 \pi
\approx 30.54 \text{ (using } \pi \approx 3.1416)
$$
✔ Area ≈ 30.54
---
🔹 Problem 2:
- Angle: $40^\circ$
- Radius: $5$
$$
\text{Area} = \frac{40}{360} \times \pi \times 5^2 = \frac{1}{9} \times \pi \times 25 = \frac{25}{9} \pi \approx 2.777 \pi
\approx 8.727
$$
✔ Area ≈ 8.73
---
🔹 Problem 3:
- Angle: $55^\circ$
- Radius: $8$
$$
\text{Area} = \frac{55}{360} \times \pi \times 8^2 = \frac{55}{360} \times \pi \times 64
= \frac{3520}{360} \pi \approx 9.778 \pi \approx 30.71
$$
✔ Area ≈ 30.71
---
🔹 Problem 4:
- Angle: $12^\circ$
- Radius: $12$
$$
\text{Area} = \frac{12}{360} \times \pi \times 12^2 = \frac{1}{30} \times \pi \times 144 = \frac{144}{30} \pi = 4.8 \pi \approx 15.08
$$
✔ Area ≈ 15.08
---
🔹 Problem 5:
- Angle: $70^\circ$
- Radius: $6$
$$
\text{Area} = \frac{70}{360} \times \pi \times 6^2 = \frac{7}{36} \times \pi \times 36 = 7\pi \approx 21.99
$$
✔ Area ≈ 21.99
---
🔹 Problem 6:
- Angle: $5^\circ$
- Radius: $9$
$$
\text{Area} = \frac{5}{360} \times \pi \times 9^2 = \frac{1}{72} \times \pi \times 81 = \frac{81}{72} \pi = 1.125 \pi \approx 3.53
$$
✔ Area ≈ 3.53
---
✔ Final Answers (rounded to two decimal places):
| Sector | Area |
|--------|------|
| 1 |
30.54 |
| 2 |
8.73 |
| 3 |
30.71 |
| 4 |
15.08 |
| 5 |
21.99 |
| 6 |
3.53 |
> 💡 Note: You can leave answers in terms of $\pi$ if preferred, but since the problem asks for "area", decimal approximations are standard unless otherwise specified.
Let me know if you'd like the exact values in terms of $\pi$!
Parent Tip: Review the logic above to help your child master the concept of area of a sector worksheet.
