Q: Given a circle with radius $$r = 8 cm$$, what is the length of the arc subtended by a central angle of $$45^ ext{o}$$? Use the formula $$s = r \theta$$ where $$\theta is in $$radians.

$$s = 8 \times \frac{45 \pi}{180} = 2\pi cm$$

Q: A car travels along a circular track of radius $$50$$ meters, subtending a central angle of $$2$$ radians. What is the distance traveled by the car?

$$s = 50 \times 2 = 100$$ meters

Q: If the radius of a circle is $$12 cm $$and the arc length is $$18 cm$$, what is the measure of the central angle in radians?

$$\theta = \frac{18}{12} = 1.5$$ radians

Q: A water sprinkler sprays water over a distance of $$20$$ feet while rotating through an angle of $$120^ ext{o}. $$What area of lawn is watered? Use $$A = \frac{1}{2} r^2$ \theta$$ with $$\theta in $$radians.

$$A = \frac{1}{2} \times 20^2$ \times \frac{2\pi}{3} = \frac{400\pi}{3} sq ft$$

Q: Which of the following is the correct formula for the area of a sector of a circle with radius $$r$$ and central angle $$\theta (in $$radians)?

$$A = \frac{1}{2} r^2$ \theta$$

Q: A sector of a circle has an area of $$30 cm$^2$$ and a radius of $$6 cm. $$What is the measure of the central angle in radians?

$$\theta = \frac{2 \times 30}{6^2$} = \frac{60}{36} = 1.67$$ radians

Q: A circular pizza has a radius of $$10$$ inches. If a slice makes a central angle of $$60^ ext{o}$$, what is the area of the slice?

$$A = \frac{1}{2} \times 10^2$ \times \frac{\pi}{3} = 50\pi/3 sq in$$

Q: A runner completes one lap around a circular track of radius $$40$$ meters. What is the distance covered?

$$s = 2\pi \times 40 = 80\pi$$ meters

Q: A clock’s minute hand is $$15 cm $$long. How far does the tip of the minute hand travel in $$20$$ minutes?

$$s = 15 \times \frac{2\pi}{3} = 10\pi cm$$

Question 1

Given a circle with radius $$r = 8 cm$$, what is the length of the arc subtended by a central angle of $$45^	ext{o}$$? Use the formula $$s = r 	heta$$ where $$	heta is in $$radians.

Accepted Answer

$$s = 8 	imes \frac{45 \pi}{180} = 2\pi cm$$

Question 2

Convert $$120^	ext{o} to $$radians.

Accepted Answer

$$\frac{2\pi}{3}$$ radians

Question 3

A car travels along a circular track of radius $$50$$ meters, subtending a central angle of $$2$$ radians. What is the distance traveled by the car?

Accepted Answer

$$s = 50 	imes 2 = 100$$ meters

Question 4

If the radius of a circle is $$12 cm $$and the arc length is $$18 cm$$, what is the measure of the central angle in radians?

Accepted Answer

$$	heta = \frac{18}{12} = 1.5$$ radians

Question 5

A water sprinkler sprays water over a distance of $$20$$ feet while rotating through an angle of $$120^	ext{o}. $$What area of lawn is watered? Use $$A = \frac{1}{2} r^2$ 	heta$$ with $$	heta in $$radians.

Accepted Answer

$$A = \frac{1}{2} 	imes 20^2$ 	imes \frac{2\pi}{3} = \frac{400\pi}{3} sq ft$$

Question 6

Which of the following is the correct formula for the area of a sector of a circle with radius $$r$$ and central angle $$	heta (in $$radians)?

Accepted Answer

$$A = \frac{1}{2} r^2$ 	heta$$

Question 7

A sector of a circle has an area of $$30 cm$^2$$ and a radius of $$6 cm. $$What is the measure of the central angle in radians?

Accepted Answer

$$	heta = \frac{2 	imes 30}{6^2$} = \frac{60}{36} = 1.67$$ radians

Question 8

If a central angle of $$\frac{\pi}{4}$$ radians subtends an arc of $$5 cm$$, what is the radius of the circle?

Accepted Answer

$$r = \frac{5}{\frac{\pi}{4}} = \frac{20}{\pi} cm$$

Question 9

A circular pizza has a radius of $$10$$ inches. If a slice makes a central angle of $$60^	ext{o}$$, what is the area of the slice?

Accepted Answer

$$A = \frac{1}{2} 	imes 10^2$ 	imes \frac{\pi}{3} = 50\pi/3 sq in$$

Question 10

A runner completes one lap around a circular track of radius $$40$$ meters. What is the distance covered?

Accepted Answer

$$s = 2\pi 	imes 40 = 80\pi$$ meters

Question 11

A central angle of $$270^	ext{o} is $$how many radians?

Accepted Answer

$$\frac{3\pi}{2}$$ radians

Question 12

A clock’s minute hand is $$15 cm $$long. How far does the tip of the minute hand travel in $$20$$ minutes?

Accepted Answer

$$s = 15 	imes \frac{2\pi}{3} = 10\pi cm$$

Question 13

A circular garden has a radius of $$25$$ meters. What is the area of a sector with a central angle of $$90^	ext{o}$$?

Accepted Answer

$$A = \frac{1}{2} 	imes 25^2$ 	imes \frac{\pi}{2} = 312.5\pi sq m$$

Question 14

If the arc length of a sector is $$14 cm $$and the radius is $$7 cm$$, what is the measure of the central angle in radians?

Accepted Answer

$$	heta = \frac{14}{7} = 2$$ radians

Question 15

A central angle of $$1.2$$ radians subtends an arc of $$9.6$$ meters. What is the radius of the circle?

Accepted Answer

$$r = \frac{9.6}{1.2} = 8$$ meters

Question 16

A circular track has a radius of $$60$$ meters. What is the area enclosed by the track?

Accepted Answer

$$A = \pi 	imes 60^2$ = 3600\pi sq m$$

Measuring Angles and Arc Lengths in Trigonometry

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