Radians, Arc Length & Angular Speed

You’ll be able to

Convert between degrees and radians and explain why radians are dimensionless
Compute arc length and sector area from a central angle in radians
Relate angular speed to linear speed on a rotating object

An angle measured by the arc it cuts

A radian is the angle subtending an arc equal in length to the radius. Since the full circumference is 2πr, a full turn is 2π radians — a number that comes from the geometry rather than from a convention. Degrees are arbitrary (Babylonian base 60); radians are not, and that is why every calculus formula for trigonometric functions assumes them.

Radian conversions and arc length

π radians = 180° · s = rθ · A = ½r²θ (θ in radians)

Multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees. Both arc-length formulas fail outright if θ is in degrees.

Why the formulas are so simple in radians

Because θ = s/r, the angle is a ratio of two lengths and therefore carries no units at all. So s = rθ needs no conversion constant, and neither does A = ½r²θ. In degrees the same formulas need a factor of π/180 bolted on. Radian measure is what buys the clean formulas, and it is the whole reason for the convention.

Worked example

A circle has radius 12 cm. A central angle of 150° cuts an arc. Find the arc length and the sector area.

1.Convert: 150° × π/180 = 5π/6 radians ≈ 2.618.
2.Arc length: s = rθ = 12(5π/6) = 10π ≈ 31.4 cm.
3.Sector area: A = ½r²θ = ½(144)(5π/6) = 72(5π/6) = 60π ≈ 188.5 cm².
4.Check: 150° is 150/360 = 5/12 of the circle. Circumference is 24π, and 5/12 of that is 10π. ✓

Answer: Arc length 10π ≈ 31.4 cm, sector area 60π ≈ 188.5 cm². The fraction-of-the-circle check is worth doing every time — it catches degree/radian confusion instantly.

Watch out

Check your calculator's angle mode before every trigonometric computation. sin(30) is 0.5 in degree mode and −0.988 in radian mode. Neither is wrong; only one answers the question you asked.

Checkpoint

A wheel of radius 0.35 m rolls without slipping through an angle of 4.2 radians. How far does it travel?

Angular speed and linear speed

Angular speed ω is angle per unit time (radians per second); linear speed v is distance per unit time. Differentiating s = rθ with respect to time — or simply dividing both sides by t — gives v = rω. So on a rigid rotating object every point shares the same angular speed but the linear speed grows with distance from the axis: the tip of a propeller moves far faster than a point near the hub, though both complete a revolution in the same time.

Angular and linear speed

ω = θ/t · v = rω · one revolution = 2π radians

Convert revolutions per minute to radians per second by multiplying by 2π/60. ω must be in radians per unit time for v = rω to hold.

Worked example

A bicycle wheel of radius 34 cm spins at 165 rpm. Find the angular speed in rad/s and the bicycle's speed in km/h.

1.ω = 165 rev/min × 2π rad/rev × 1 min/60 s = 165(2π)/60 ≈ 17.28 rad/s.
2.v = rω = 0.34 m × 17.28 rad/s ≈ 5.875 m/s.
3.Convert: 5.875 m/s × 3600 s/h = 21,150 m/h.
4.That is about 21.2 km/h.

Answer: ω ≈ 17.3 rad/s and the bicycle travels about 21.2 km/h. Converting the radius to meters before computing v keeps the units consistent — mixing centimeters with meters per second is the usual source of a factor-of-100 error here.

Checkpoint

Two points on a spinning disk sit at radii 5 cm and 15 cm. How do their speeds compare?

Answer the 2 checkpoints as you read.