Practice worksheet for calculating radius, central angle, and arc length in circles.

Here are the step-by-step solutions for each problem on the worksheet. We will use the formula provided in the image:
Arc Length ($s$) = $\frac{\text{central angle}}{360} \times 2 \times \pi \times \text{radius}$

We will use $\pi \approx 3.14$.

Problem 1

* Given: Radius ($r$) = $15$ ft, Central Angle ($\theta$) = $210^\circ$
* Find: Arc Length ($s$)
* Calculation:
$$s = \frac{210}{360} \times 2 \times 3.14 \times 15$$
$$s = 0.5833 \times 6.28 \times 15$$
$$s = 54.95$$
* Answers:
* Radius = 15 ft
* Central angle = 210°
* Length of the arc PQ = 54.95 ft

Problem 2

* Given: Arc Length ($s$) = $15.87$ m, Central Angle ($\theta$) = $78^\circ$
* Find: Radius ($r$)
* Calculation:
Rearrange the formula to solve for $r$: $r = \frac{s \times 360}{\theta \times 2 \times \pi}$
$$r = \frac{15.87 \times 360}{78 \times 2 \times 3.14}$$
$$r = \frac{5713.2}{489.84}$$
$$r = 11.663...$$
Rounding to the nearest whole number: $12$
* Answers:
* Radius = 12 m
* Central angle = 78°
* Length of the arc AB = 15.87 m

Problem 3

* Given: Arc Length ($s$) = $9.6$ yd, Central Angle ($\theta$) = $58^\circ$
* Find: Radius ($r$)
* Calculation:
$$r = \frac{9.6 \times 360}{58 \times 2 \times 3.14}$$
$$r = \frac{3456}{364.24}$$
$$r = 9.488...$$
Rounding to the nearest whole number: $9$
* Answers:
* Radius = 9 yd
* Central angle = 58°
* Length of the arc EF = 9.6 yd

Problem 4

* Given: Radius ($r$) = $13$ dt, Central Angle ($\theta$) = $225^\circ$
* Find: Arc Length ($s$)
* Calculation:
$$s = \frac{225}{360} \times 2 \times 3.14 \times 13$$
$$s = 0.625 \times 6.28 \times 13$$
$$s = 51.025$$
Rounding to two decimal places: $51.03$
* Answers:
* Radius = 13 dt
* Central angle = 225°
* Length of the arc RS = 51.03 dt

Problem 5

* Given: Arc Length ($s$) = $59.16$ in, Central Angle ($\theta$) = $90^\circ$
* Find: Radius ($r$)
* Calculation:
$$r = \frac{59.16 \times 360}{90 \times 2 \times 3.14}$$
$$r = \frac{21297.6}{565.2}$$
$$r = 37.68...$$
Rounding to the nearest whole number: $38$
* Answers:
* Radius = 38 in
* Central angle = 90°
* Length of the arc CD = 59.16 in

Problem 6

* Given: Radius ($r$) = $18$ ft, Central Angle ($\theta$) = $142^\circ$
* Find: Arc Length ($s$)
* Calculation:
$$s = \frac{142}{360} \times 2 \times 3.14 \times 18$$
$$s = 0.3944 \times 6.28 \times 18$$
$$s = 44.586...$$
Rounding to two decimal places: $44.59$
* Answers:
* Radius = 18 ft
* Central angle = 142°
* Length of the arc GH = 44.59 ft

Problem 7

* Given: Arc Length ($s$) = $54.91$ ft, Radius ($r$) = $31$ ft
* Find: Central Angle ($\theta$)
* Calculation:
Rearrange formula for $\theta$: $\theta = \frac{s \times 360}{2 \times \pi \times r}$
$$\theta = \frac{54.91 \times 360}{2 \times 3.14 \times 31}$$
$$\theta = \frac{19767.6}{194.68}$$
$$\theta = 101.53...$$
Rounding to the nearest whole number: $102$
* Answers:
* Radius = 31 ft
* Central angle = 102°
* Length of the arc KL = 54.91 ft

Problem 8

* Given: Radius ($r$) = $14$ ft, Central Angle ($\theta$) = $120^\circ$
* Find: Arc Length ($s$)
* Calculation:
$$s = \frac{120}{360} \times 2 \times 3.14 \times 14$$
$$s = \frac{1}{3} \times 6.28 \times 14$$
$$s = 29.306...$$
Rounding to two decimal places: $29.31$
* Answers:
* Radius = 14 ft
* Central angle = 120°
* Length of the arc XY = 29.31 ft

Problem 9

* Given: Arc Length ($s$) = $13.64$ m, Central Angle ($\theta$) = $148^\circ$
* Find: Radius ($r$)
* Calculation:
$$r = \frac{13.64 \times 360}{148 \times 2 \times 3.14}$$
$$r = \frac{4910.4}{929.44}$$
$$r = 5.28...$$
Rounding to the nearest whole number: $5$
* Answers:
* Radius = 5 m
* Central angle = 148°
* Length of the arc PQ = 13.64 m

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Final Answer:

1) Radius = 15 ft, Central angle = 210°, Arc length = 54.95 ft
2) Radius = 12 m, Central angle = 78°, Arc length = 15.87 m
3) Radius = 9 yd, Central angle = 58°, Arc length = 9.6 yd
4) Radius = 13 dt, Central angle = 225°, Arc length = 51.03 dt
5) Radius = 38 in, Central angle = 90°, Arc length = 59.16 in
6) Radius = 18 ft, Central angle = 142°, Arc length = 44.59 ft
7) Radius = 31 ft, Central angle = 102°, Arc length = 54.91 ft
8) Radius = 14 ft, Central angle = 120°, Arc length = 29.31 ft
9) Radius = 5 m, Central angle = 148°, Arc length = 13.64 m