Tuesday, February 8, 2011


Trigonometry students have this to look forward to, but those in College Algebra may be reexamining these issues right now. Although linear and angular velocity questions can be answered using Geometry concepts alone, calculating circumference and using degrees can create long solutions with lots of opportunities to make silly, little mistakes. Here are the equations you need in order to turn these problems into simple substitution work.

First, take a look at the names of these concepts: LINEAR deals with lines and is the familiar “miles per hour” measurement. In circular motion, it's the line measure we call circumference. ANGULAR deals with angles in a circle -- in common terms RPM, or revolutions per minute. Velocity, as we commonly know it, is distance divided by time.

To make things simple, we can calculate Angular Velocity first and use it to quickly find Linear Velocity.

Remember in Geometry that we calculated the arc length.

Given a radius of 4 and a central angle of 50... S = (50/360)• 4•4π = 20π/9.

From Trig, we can convert the degrees to radians:

That’s your first equation...

To find Angular Velocity, we want the central angle measured in radians, θ, divided by time. Oh, and we get a new, cute symbol that some will be calling “W,” but what is actually the Greek letter “Omega,” ω.

That’s your second equation...
If the question gives you RPM (revolutions per minute), you can convert rpm to radians by multiplying by 2π, the radians in each time around the circle. Don't forget to convert minutes if a different time measurement is requested.

So starting with the old velocity equation, substituting S (arc length) for distance, then substituting r θ for S, and finally substituting ω (omega) for θ over t , we have the final equation: Linear Velocity.

If equations are difficult for you to remember, try printing up this visual display:

Linear velocity is the outside equation; angular velocity is the inside equation.


  1. Nice work. Thank you.
    Tony Sudarno

  2. wondering why arc length in example isn't 10pi/9?

  3. Captain, typo....thanks for the heads up! Now arithmetic is correct...
    Given a radius of 4 and a central angle of 50... S = (50/360)• 4•4π = 20π/9.

    You're attention to detail is impressive. Of nearly 9,000 views of that calculation, you're the FIRST AND ONLY to have checked it out. Nice Job, Captain....SF

  4. Isn't it 10pi/9?

    given a central angle of 50 degrees and a radius of 4,

    50 degrees is 50 * (pi/180 degrees)= (5pi)/18

    arc length(s)=radius(r) * central angle(theta in radians)

    s = 4 * ((5pi)/18) = (20pi)/18 = (10pi)/9?

  5. I see it....angle/360 = 50/360
    times 2πr = 2•4•π
    copy reads 4•4•π...your edit is perfect..and I'm running to Walgreens immediately to buy a new pair of reading glasses. ;>)