Question 1

Given a circle with radius $$r = 8 cm $$and a central angle of $$	heta = \frac{\pi}{3}$$ radians, what is the length of the arc subtended by the angle?

Accepted Answer

$$\frac{8\pi}{3} cm$$

Question 2

A sector of a circle has a radius of $$10 m $$and a central angle of $$60^\circ. $$What is the area of the sector?

Accepted Answer

$$\frac{100\pi}{6}$$ $$m^2$$

Question 3

If a car tire with a radius of $$0.35 m $$rotates at $$120$$ revolutions per minute, what is the linear speed of a point on the edge of the tire?

Accepted Answer

$$264 m/$$min

Question 4

Convert an angular speed of $$30\pi$$ radians per second to revolutions per minute.

Accepted Answer

$$900$$ rev/min

Question 5

A sector of a circle has an area of $$20\pi cm$^2$$ and a radius of $$10 cm. $$What is the measure of the central angle in radians?

Accepted Answer

$$2$$ radians

Question 6

A Ferris wheel with a radius of $$15 m $$completes one revolution every $$40$$ seconds. What is the linear speed of a seat on the edge?

Accepted Answer

$$\frac{3\pi}{4} m/s$$

Question 7

If the radius of the Earth is approximately $$6371 km$$, what is the distance between two cities on the same latitude separated by a central angle of $$0.5$$ radians?

Accepted Answer

$$3185.5 km$$

Question 8

A runner is moving along a circular track with a radius of $$20 m at an $$angular speed of $$0.4$$ radians per second. What is the runner's linear speed?

Accepted Answer

$$8 m/s$$

Question 9

A sector of a circle has a central angle of $$\frac{\pi}{4}$$ radians and an area of $$8 cm$^2. $$What is the radius of the circle?

Accepted Answer

$$4 cm$$

Question 10

A rotating disk has an angular speed of $$10$$ radians per second. If the radius of the disk is $$0.5 m$$, what is the linear speed at the edge?

Accepted Answer

$$5 m/s$$

Question 11

A sector of a circle has a radius of $$12 cm $$and a central angle of $$150^\circ. $$What is the length of the arc?

Accepted Answer

$$10\pi cm$$

Question 12

A gear rotates at $$45$$ revolutions per minute. What is its angular speed in radians per second?

Accepted Answer

$$4.71$$ rad/s

Question 13

A circular pizza has a radius of $$14$$ inches. If a slice has a central angle of $$\frac{\pi}{6}$$ radians, what is the area of the slice?

Accepted Answer

$$\frac{49\pi}{3}$$ $$in^2$$

Question 14

A satellite orbits Earth at a radius of $$7000 km $$and completes one revolution every $$2$$ hours. What is its linear speed?

Accepted Answer

$$6.11 km/$$min

Question 15

A circular track has a circumference of $$400 m. If a $$runner completes $$3$$ laps in $$10$$ minutes, what is the runner's average linear speed?

Accepted Answer

$$2 m/s$$

Question 16

A rotating wheel has a radius of $$0.75 m $$and an angular speed of $$8$$ radians per second. What is the linear speed at the rim?

Accepted Answer

$$6 m/s$$

Question 17

A sector of a circle has an area of $$25 cm$^2$$ and a central angle of $$\frac{\pi}{5}$$ radians. What is the radius of the circle?

Accepted Answer

$$5 cm$$

Question 18

A merry-go-round has a radius of $$4 m $$and rotates at $$0.5$$ revolutions per second. What is the linear speed of a point on the edge?

Accepted Answer

$$4\pi m/s$$

Question 19

A circular garden has a radius of $$6 m. $$What is the length of the fence needed to enclose a sector with a central angle of $$120^\circ$$?

Accepted Answer

$$12 + 4\pi m$$

Question 20

A clock's minute hand is $$0.2 m $$long. What is the linear speed of the tip of the hand?

Accepted Answer

$$0.021 m/s$$

Arc Length, Sector Area, and Angular/Linear Speed in Trigonometry

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