It's easy to understand the formula for the twist angle in a section due to torsion, and all the units neatly cancel out leaving a unitless number.

It might sound like a very simple question that it's stupid, I mean the result is very intuitive but I still can't make out how people know that the result would be in radians.

I reckon it has something to do with torque in the formula since it's the only variable with something to do with circles but I can't figure out the relationship.

  • 1
    $\begingroup$ If you need to differentiate anything with trigonometric functions then you must use radians or then the differentiation will not work out. Now since twist angle is a integral of the forces and torques acting on its body. So naturally youd end up with radians. $\endgroup$
    – joojaa
    Oct 21, 2018 at 19:52
  • $\begingroup$ ^this is the crux of the answer imho. $\endgroup$ Oct 21, 2018 at 20:40

2 Answers 2


The results of twist angle is indeed dimensionless, but so are radians!

The radian is defined as the ratio of arc length to radius length, so both arc length and radius length have the same unit, then they cancel each other.

It's common for many dimensionless quantities in engineering and physics to use radians. You'll get used to it.

  • $\begingroup$ A radian is defined as 57.3 degrees, so what other dimensionless quantities are called radians? $\endgroup$
    – Solar Mike
    Oct 21, 2018 at 13:21
  • $\begingroup$ All angles are dimensionless, but that doesn't mean they don't have units... math.stackexchange.com/questions/1097581/… $\endgroup$ Oct 21, 2018 at 13:38
  • $\begingroup$ Radians is a unit that has no base dimensions (length, time, etc.). $\endgroup$ Oct 21, 2018 at 13:40
  • $\begingroup$ The Shockley exponent for example is defined in radians, it's a physical quantity $\endgroup$
    – user14407
    Oct 21, 2018 at 14:01
  • $\begingroup$ So it is defined with radians, does not mean it is called a radian... Perhaps where you put "to refer to many dimensionless quantities in engineering and physics as radians" you actually meant "many dimensionless quantities in engineering and physics use radians" $\endgroup$
    – Solar Mike
    Oct 21, 2018 at 14:24

Get back to basics, and the definition of shear strain. (Image source: http://www.materials.unsw.edu.au/tutorials/online-tutorials/5-shear-strain)

enter image description here

Unless you want to clutter up your formulas with factors of $\pi/180$, measuring $\theta$ in radians gives the shear strain $\gamma$ as $$\gamma = \frac w L = \tan \theta \approx \theta.$$


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