Arc Length and Sector Area Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem step by step using the formulas for arc length and sector area.

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🔷 Key Formulas:

- Arc Length:
$$
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
$$

- Sector Area:
$$
\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2
$$

Where:
- $\theta$ is the central angle in degrees,
- $r$ is the radius of the circle.

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✔ Problem 1: Find the length of the bolded arc.

Given: Radius = 3 mm, Central angle = 108°

$$
\text{Arc Length} = \frac{108}{360} \times 2\pi(3) = \frac{3}{10} \times 6\pi = 1.8\pi \approx 5.65 \text{ mm}
$$

✔ Answer: $1.8\pi$ mm or approximately 5.65 mm

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✔ Problem 2: Find the area of the shaded sector.

Given: Radius = 8 ft, Central angle = 32°

$$
\text{Area} = \frac{32}{360} \times \pi(8)^2 = \frac{4}{45} \times 64\pi = \frac{256}{45}\pi \approx 18.07 \text{ ft}^2
$$

✔ Answer: $\frac{256}{45}\pi$ ft² or approximately 18.07 ft²

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✔ Problem 3: Find the length of the bolded arc.

Given: Radius = 6 in, Central angle = 143°

$$
\text{Arc Length} = \frac{143}{360} \times 2\pi(6) = \frac{143}{360} \times 12\pi = \frac{1716}{360}\pi = \frac{143}{30}\pi \approx 14.96 \text{ in}
$$

✔ Answer: $\frac{143}{30}\pi$ in or approximately 14.96 in

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✔ Problem 4: Find the area of the shaded sector.

Given: Radius = 12 cm, Central angle = 204°

$$
\text{Area} = \frac{204}{360} \times \pi(12)^2 = \frac{17}{30} \times 144\pi = \frac{2448}{30}\pi = 81.6\pi \approx 256.36 \text{ cm}^2
$$

✔ Answer: $81.6\pi$ cm² or approximately 256.36 cm²

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✔ Problem 5: Find the length of the bolded arc.

Given: Diameter = 30 m → Radius = 15 m, Central angle = 47°

$$
\text{Arc Length} = \frac{47}{360} \times 2\pi(15) = \frac{47}{360} \times 30\pi = \frac{1410}{360}\pi = \frac{47}{12}\pi \approx 12.33 \text{ m}
$$

✔ Answer: $\frac{47}{12}\pi$ m or approximately 12.33 m

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✔ Problem 6: Find the area of the shaded sector.

Given: Diameter = 17 mi → Radius = 8.5 mi, Central angle = 126°

$$
\text{Area} = \frac{126}{360} \times \pi(8.5)^2 = \frac{7}{20} \times 72.25\pi = \frac{505.75}{20}\pi = 25.2875\pi \approx 79.44 \text{ mi}^2
$$

✔ Answer: $25.2875\pi$ mi² or approximately 79.44 mi²

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✔ Problem 7: Find the length of an arc subtended by a 70° central angle in a circle with radius 10 cm.

$$
\text{Arc Length} = \frac{70}{360} \times 2\pi(10) = \frac{7}{36} \times 20\pi = \frac{140}{36}\pi = \frac{35}{9}\pi \approx 12.22 \text{ cm}
$$

✔ Answer: $\frac{35}{9}\pi$ cm or approximately 12.22 cm

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✔ Problem 8: Find the area of a sector formed by a 105° central angle in a circle with radius 5 in.

$$
\text{Area} = \frac{105}{360} \times \pi(5)^2 = \frac{7}{24} \times 25\pi = \frac{175}{24}\pi \approx 23.03 \text{ in}^2
$$

✔ Answer: $\frac{175}{24}\pi$ in² or approximately 23.03 in²

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✔ Problem 9: What is the length of an arc created by a 280° central angle in a circle with diameter 100 m?

→ Radius = 50 m

$$
\text{Arc Length} = \frac{280}{360} \times 2\pi(50) = \frac{7}{9} \times 100\pi = \frac{700}{9}\pi \approx 244.35 \text{ m}
$$

✔ Answer: $\frac{700}{9}\pi$ m or approximately 244.35 m

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📝 Final Answers Summary:

| Problem | Answer (Exact) | Approximate |
|--------|------------------|-------------|
| 1 | $1.8\pi$ mm | 5.65 mm |
| 2 | $\frac{256}{45}\pi$ ft² | 18.07 ft² |
| 3 | $\frac{143}{30}\pi$ in | 14.96 in |
| 4 | $81.6\pi$ cm² | 256.36 cm² |
| 5 | $\frac{47}{12}\pi$ m | 12.33 m |
| 6 | $25.2875\pi$ mi² | 79.44 mi² |
| 7 | $\frac{35}{9}\pi$ cm | 12.22 cm |
| 8 | $\frac{175}{24}\pi$ in² | 23.03 in² |
| 9 | $\frac{700}{9}\pi$ m | 244.35 m |

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