Arc Length-Sector Area, Wells Worksheet (Given) PDF | PDF - Free Printable

Let's solve each problem step by step using the arc length formula:

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

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

We’ll also use:
- Circumference: $C = 2\pi r$
- Diameter: $d = 2r$

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1. Find the length of arc $AB$.

Given:
- Radius $r = 6$ cm
- Central angle $\theta = 120^\circ$

$$
\text{Arc Length} = \frac{120}{360} \times 2\pi \times 6 = \frac{1}{3} \times 12\pi = 4\pi \text{ cm}
$$

✔ Answer: $4\pi$ cm

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2. The diameter is 24 cm. Find the length of arc $CD$.

Given:
- Diameter = 24 cm → Radius $r = 12$ cm
- Central angle $\theta = 60^\circ$

$$
\text{Arc Length} = \frac{60}{360} \times 2\pi \times 12 = \frac{1}{6} \times 24\pi = 4\pi \text{ cm}
$$

✔ Answer: $4\pi$ cm

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3. The length of arc $EF$ is $5\pi$ in. Find the length of the radius.

Given:
- Arc length = $5\pi$ inches
- Central angle $\theta = 30^\circ$

Use the formula:

$$
\text{Arc Length} = \frac{\theta}{360} \times 2\pi r
$$

$$
5\pi = \frac{30}{360} \times 2\pi r = \frac{1}{12} \times 2\pi r = \frac{\pi r}{6}
$$

Now solve for $r$:

$$
5\pi = \frac{\pi r}{6} \Rightarrow 5 = \frac{r}{6} \Rightarrow r = 30 \text{ inches}
$$

✔ Answer: 30 inches

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4. Find the length of arc $XY$.

Given:
- Radius $r = 10$ cm
- Central angle $\theta = 70^\circ$

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

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

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5. A circle has an arc whose measure is $80^\circ$ and whose length is $88\pi$. What is the diameter of the circle?

Given:
- Arc length = $88\pi$
- $\theta = 80^\circ$

Use arc length formula:

$$
88\pi = \frac{80}{360} \times 2\pi r = \frac{2}{9} \times 2\pi r = \frac{4\pi r}{9}
$$

Solve for $r$:

$$
88\pi = \frac{4\pi r}{9} \Rightarrow 88 = \frac{4r}{9} \Rightarrow r = \frac{88 \times 9}{4} = 198
$$

So, radius $r = 198$, then diameter $d = 2r = 396$

✔ Answer: 396 units

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6. A circle has a circumference whose length is $25\pi$. Find the length of an arc whose central angle is $90^\circ$.

Given:
- Circumference $C = 25\pi$
- So $2\pi r = 25\pi \Rightarrow r = \frac{25}{2} = 12.5$

Now find arc length for $90^\circ$:

$$
\text{Arc Length} = \frac{90}{360} \times 2\pi r = \frac{1}{4} \times 25\pi = \frac{25\pi}{4}
$$

Alternatively, since $90^\circ$ is $1/4$ of the circle, arc length is $1/4$ of circumference:

$$
\frac{1}{4} \times 25\pi = \frac{25\pi}{4}
$$

✔ Answer: $\frac{25\pi}{4}$ units

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7. Find the measure of the central angle of an arc if its length is $14\pi$ and the radius is 18.

Given:
- Arc length = $14\pi$
- Radius $r = 18$

Use:

$$
\text{Arc Length} = \frac{\theta}{360} \times 2\pi r
$$

$$
14\pi = \frac{\theta}{360} \times 2\pi \times 18 = \frac{\theta}{360} \times 36\pi
$$

Divide both sides by $\pi$:

$$
14 = \frac{\theta}{360} \times 36 \Rightarrow 14 = \frac{36\theta}{360} = \frac{\theta}{10}
\Rightarrow \theta = 140^\circ
$$

✔ Answer: $140^\circ$

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✔ Final Answers Summary:

1. $4\pi$ cm
2. $4\pi$ cm
3. 30 inches
4. $\frac{35\pi}{9}$ cm
5. 396 units
6. $\frac{25\pi}{4}$ units
7. $140^\circ$

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