Worksheet on calculating arc length for circular sectors, with examples and practice problems.

To solve the problem of calculating the arc length for each sector, we use the formula for the arc length of a circle:

\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
\]

where:
- \(\theta\) is the central angle in degrees,
- \(r\) is the radius of the circle,
- \(\pi\) is approximately \(3.14159\).

Let's solve each part step by step.

---

Question 1

#### (a)
- Radius (\(r\)): \(6 \, \text{cm}\)
- Central angle (\(\theta\)): \(90^\circ\) (since it is a quarter-circle)

\[
\text{Arc Length} = \frac{90^\circ}{360^\circ} \times 2\pi \times 6
\]

\[
= \frac{1}{4} \times 2\pi \times 6
\]

\[
= \frac{1}{4} \times 12\pi
\]

\[
= 3\pi
\]

Using \(\pi \approx 3.14159\):

\[
3\pi \approx 3 \times 3.14159 = 9.42477
\]

Rounding to one decimal place:

\[
\boxed{9.4 \, \text{cm}}
\]

#### (b)
- Radius (\(r\)): \(15 \, \text{cm}\)
- Central angle (\(\theta\)): \(120^\circ\)

\[
\text{Arc Length} = \frac{120^\circ}{360^\circ} \times 2\pi \times 15
\]

\[
= \frac{1}{3} \times 2\pi \times 15
\]

\[
= \frac{1}{3} \times 30\pi
\]

\[
= 10\pi
\]

Using \(\pi \approx 3.14159\):

\[
10\pi \approx 10 \times 3.14159 = 31.4159
\]

Rounding to one decimal place:

\[
\boxed{31.4 \, \text{cm}}
\]

#### (c)
- Radius (\(r\)): \(4 \, \text{m}\)
- Central angle (\(\theta\)): \(30^\circ\)

\[
\text{Arc Length} = \frac{30^\circ}{360^\circ} \times 2\pi \times 4
\]

\[
= \frac{1}{12} \times 2\pi \times 4
\]

\[
= \frac{1}{12} \times 8\pi
\]

\[
= \frac{8\pi}{12}
\]

\[
= \frac{2\pi}{3}
\]

Using \(\pi \approx 3.14159\):

\[
\frac{2\pi}{3} \approx \frac{2 \times 3.14159}{3} = \frac{6.28318}{3} \approx 2.09439
\]

Rounding to one decimal place:

\[
\boxed{2.1 \, \text{m}}
\]

#### (d)
- Radius (\(r\)): \(20 \, \text{cm}\)
- Central angle (\(\theta\)): \(72^\circ\)

\[
\text{Arc Length} = \frac{72^\circ}{360^\circ} \times 2\pi \times 20
\]

\[
= \frac{1}{5} \times 2\pi \times 20
\]

\[
= \frac{1}{5} \times 40\pi
\]

\[
= 8\pi
\]

Using \(\pi \approx 3.14159\):

\[
8\pi \approx 8 \times 3.14159 = 25.13272
\]

Rounding to one decimal place:

\[
\boxed{25.1 \, \text{cm}}
\]

---

Question 2

#### (a)
- Radius (\(r\)): \(8 \, \text{cm}\)
- Central angle (\(\theta\)): \(14^\circ\)

\[
\text{Arc Length} = \frac{14^\circ}{360^\circ} \times 2\pi \times 8
\]

\[
= \frac{14}{360} \times 2\pi \times 8
\]

\[
= \frac{7}{180} \times 16\pi
\]

\[
= \frac{112\pi}{180}
\]

\[
= \frac{28\pi}{45}
\]

Using \(\pi \approx 3.14159\):

\[
\frac{28\pi}{45} \approx \frac{28 \times 3.14159}{45} = \frac{87.96452}{45} \approx 1.954767
\]

Rounding to one decimal place:

\[
\boxed{2.0 \, \text{cm}}
\]

#### (b)
- Radius (\(r\)): \(19 \, \text{cm}\)
- Central angle (\(\theta\)): \(150^\circ\)

\[
\text{Arc Length} = \frac{150^\circ}{360^\circ} \times 2\pi \times 19
\]

\[
= \frac{5}{12} \times 2\pi \times 19
\]

\[
= \frac{5}{12} \times 38\pi
\]

\[
= \frac{190\pi}{12}
\]

\[
= \frac{95\pi}{6}
\]

Using \(\pi \approx 3.14159\):

\[
\frac{95\pi}{6} \approx \frac{95 \times 3.14159}{6} = \frac{298.49605}{6} \approx 49.74934
\]

Rounding to one decimal place:

\[
\boxed{49.7 \, \text{cm}}
\]

#### (c)
- Radius (\(r\)): \(25.2 \, \text{cm}\)
- Central angle (\(\theta\)): \(73^\circ\)

\[
\text{Arc Length} = \frac{73^\circ}{360^\circ} \times 2\pi \times 25.2
\]

\[
= \frac{73}{360} \times 2\pi \times 25.2
\]

\[
= \frac{73}{360} \times 50.4\pi
\]

\[
= \frac{3688.8\pi}{360}
\]

\[
= \frac{922.2\pi}{90}
\]

Using \(\pi \approx 3.14159\):

\[
\frac{922.2\pi}{90} \approx \frac{922.2 \times 3.14159}{90} = \frac{2896.777398}{90} \approx 32.1864155
\]

Rounding to one decimal place:

\[
\boxed{32.2 \, \text{cm}}
\]

#### (d)
- Radius (\(r\)): \(60 \, \text{cm}\)
- Central angle (\(\theta\)): \(235^\circ\)

\[
\text{Arc Length} = \frac{235^\circ}{360^\circ} \times 2\pi \times 60
\]

\[
= \frac{235}{360} \times 2\pi \times 60
\]

\[
= \frac{235}{360} \times 120\pi
\]

\[
= \frac{28200\pi}{360}
\]

\[
= \frac{282\pi}{3}
\]

Using \(\pi \approx 3.14159\):

\[
\frac{282\pi}{3} \approx \frac{282 \times 3.14159}{3} = \frac{885.77058}{3} \approx 295.25686
\]

Rounding to one decimal place:

\[
\boxed{295.3 \, \text{cm}}
\]

---

Final Answers:

1. (a) \(\boxed{9.4 \, \text{cm}}\)
(b) \(\boxed{31.4 \, \text{cm}}\)
(c) \(\boxed{2.1 \, \text{m}}\)
(d) \(\boxed{25.1 \, \text{cm}}\)

2. (a) \(\boxed{2.0 \, \text{cm}}\)
(b) \(\boxed{49.7 \, \text{cm}}\)
(c) \(\boxed{32.2 \, \text{cm}}\)
(d) \(\boxed{295.3 \, \text{cm}}\)