Suppose [tex]a,b,c[/tex] are measured in meters. Then the semiperimeter [tex]s=\dfrac{a+b+c}2[/tex] is also given in meters, so [tex]s-a,s-b,s-c[/tex] are all also given in meters. When multiplied together, [tex](s-a)(s-b)(s-c)[/tex] is measured in cubic meters. Divide this by [tex]s[/tex], measured in meters, and this reduces to square meters. Take the square root and you end up back with meters once again. So the formula is indeed dimensionally consistent.
