Cogito
Trigonometry · Chapter 2 · Lesson 3
Arc Length and Angular Speed
What radians were actually invented for.
12 problems · about 21 minutes · F-TF.A.1, G-C.B.5
What this lesson teaches
The student computes arc length, sector area and angular speed using radian measure.
- Arc length is s = rθ and sector area is ½r²θ, with θ in radians.
- Angular speed is radians per second; linear speed is v = rω.
- Every point on a wheel shares an angular speed but not a linear one.
Warm Up
Straightforward practice. Get the method working first.
5 problemsRadius 7, angle 2 radians. What is the arc length?
Answer 14
Why 14.
Why must θ be in radians for s = rθ?
Answer A radian is defined as one radius of arc, so the formula needs no extra factor.
Why The definition of a radian is what removes the conversion factor.
Radius 8, angle 2 radians. Arc length?
Answer 16
Why rθ.
Radius 4, angle 3 radians. Sector area?
Answer 24
Why ½ × 16 × 3.
Arc 20, radius 5. What is the angle in radians?
Answer 4
Why θ = s ÷ r.
Build It Up
The same ideas with more to keep track of.
3 problemsRadius 3 m, angular speed 5 rad/s. Linear speed in m/s?
Answer 15
Why v = rω.
How many radians is one full turn, to two decimal places?
Answer 6.28
Why 2π.
A wheel turns 10 times per second. Angular speed in radians per second, to one decimal place?
Answer 62.8
Why 10 × 2π.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsTwo points sit on the same spinning wheel, one near the hub and one at the rim. Which statements are true?
Answer They share the same angular speed.; The rim point has the greater linear speed.
Why One turn takes the same time for both, but the rim covers more distance.
Sector area 50, radius 10. Angle in radians?
Answer 1
Why 50 = ½ × 100 × θ.
The Arc: Radius 5, angle 3 radians. What is the arc length?
Answer 15
Why 15.
The Wheel: A wheel of radius 0.5 m spins at 4 radians per second. What is the rim speed, in metres per second?
Answer 2 m/s
Why 2 m/s.