Why is length often a unitless quantity in mathematics? This is something that's bothered me for a long time: Why is length often not given any units (e.g. inches or meters) in mathematics? The unit circle is said to have a "radius of 1," with no unit given. A vector is said to have a magnitude of, say, 27, with no unit given. A triangle is said to have a hypotenuse length of, say, 10, again with no unit given. 
Is the reason no unit is given simply because one could substitute any unit, so long as the proportions between quantities remain the same? I searched around for a little on here but couldn't find any questions directly addressing my concern here. I'd appreciate any clarification. 

Ideally, you should be using units and most practical-based questions do in fact have units assigned to the variables.
But yes, when doing geometry or calculus the units are ignored because it is not required and any unit will suffice. If its really bothering you, you may just add $\text{units}$ to the end of the length, $\text{units}^2$ to the end of an area quantity and so on. I do this all the time and no teacher will mind.
