Equation Editor Syntax - Arc length = radius * angle

matthew_stacy

I'm having trouble entering a formula into a sketched length dimension.  The intent is to set the length of the construction line segment equal the length of the adjacent arc, from variables #R and #theta (millimeters and degrees, respectively).  The correct result should be approximately 34.9mm (same as the arc length).

The formula should be the product of radius and swept angle (with the angle expressed in radians not degrees):

• #R*#theta*PI/180,           (multiplying by π radians per 180 degrees to convert the angle from degrees to radians)
  

I've tried numerous variants, with and without the PI/180 conversion, peppered with rad and deg in every conceivable location.  Could someone please provide the correct syntax for this expression?  So far I keep running into the Enter a valid expression prompt.

Here is a link to the document:  https://cad.onshape.com/documents/ba77da0d35372c64be7be8bf/w/10b3901e704eabe14945dea2/e/482389f1f55cb0146b53c145

NeilCooke

You need to get the units to balance. The result of #R*#theta*PI/180 is length*angle so you need to get rid of the angle. Try #R*#theta*PI/(180*degree) 

matthew_stacy

That was just the ticket.  Thank you @NeilCooke .

ilya_baran

Even easier is #R*#theta/radian

matthew_stacy

@ilya_baran, that works,  but please explain why.  What does /radian mean?

From the perspective of dimensional analysis I've got the product of length (mm) and angle (degrees), but need to convert the angular unit to radians.  Is /radian code for a function call to "convert #theta from degrees to radians"?  That is not intuitive to me.

ilya_baran

The way to think about this is that #theta is an angle, rather than some number of degrees or radians (how we store it internally is beside the point).  In other words, #theta doesn't remember what units it was specified in, just what the actual angle was.  radian is a constant that is also an angle (that measures 1 radian, obviously).  So if you divide #theta/radian, you're dividing two angles and what you should get is a number.  That is the number of radians in #theta.

Similarly, if you have a length and need to know how many millimeters it is, you would do #myLength / millimeter.  For example if #myLength is 3mm, you would get: #myLength / millimeter = 3 * millimeter / millimeter = 3.

Hope this helps.

matthew_stacy

ilya_baran, "how we store it internally is beside the point", but you have surely piqued my interest.  If I listened to the little man in the wizard of Oz saying, "don't worry about what's behind the curtain", I probably wouldn't have studied engineering all those years ago.  How does Onshape store an angular dimension internally?

To verify your assertion, I created a new variable #num = #theta/rad

Sure enough, if I set the value of #theta to 180degrees the value of #num updates to 3.142 which looks to be a perfectly good approximation of Pi as you promised.  Or if I set #num = #theta/deg the value updates to 180.

This is fabulous, but what's under the hood?

ilya_baran

It's a lot more boring than I led you to believe   Internally, we just convert everything to radians, (or meters, for lengths) and do all computations in those "base" units.  But ideally, we could change the base units to hexacontades and smoots tomorrow and nothing would break (we're of course not going to do that).

matthew_stacy

Thanks @ilya_baran