Best Answer:
The previous responder is actually answering a DIFFERENT question, namely one about the AREA of a triangle, with dimensions L^2!

In YOUR question, the LHS (r) is a RADIUS, and thus has the dimension of a (single) length, that is to say L.

Meanwhile the RHS, properly written as √[(s-a)(s-b)(s-c)/s], to make it clear that the ' √ ' sign operates on EVERYTHING to its right, has dimension(s) of √[L*L*L / L] = √[L^2] = L.

So the equation is dimensionally consistent.

THAT'S what you wanted to know. QED

Live long and prosper.

P.S. Do NOT make the mistake of confusing the question of the DIMENSIONALITY of something (the combinations of the different dimensional essences L, M and T involved in it) with the UNITS that you choose to assign to those dimensionalities.

This seems to be a very common error in YA! answers, which may well represent widespread confusion among today's teachers. One of the main reasons for using dimensional arguments is that they get you away from using a SPECIFIC choice of units! The results that you obtain from a dimensional argument will be true in ANY unit system. And from simply a writing point of view, it is so much more EFFICIENT, CLEANER and CLEARER to write 'L' instead of 'meters' or M instead of 'kg' etc. Just notice how the single CAPITAL letters jump out at your eye instead of all the clutter of multiple, repeated lower case letters.

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