Why is the Heron's Formula taught in a different way? In high school, I was taught Heron's Formula in this format: let $s$ denote semiperimeter of $∆ABC$ then $$[∆ABC] = \sqrt{s(s-a)(s-b)(s-c)}$$ where the symbols have their usual meanings. I feel some 'awkward' in this formula. Firstly, it introduces a new variable $s$ though I can easily calculate the value of it. Secondly, it is efficient only with rational values. When we have irrational values, solving using conventional method might be too lengthy. For example, take $a = 5$, $b = \sqrt{73}$, $c = \sqrt{52}$. We cannot calculate the area in one line. (In my case, I encountered these values while solving an integer answer type question and seeing the 'roots' made me question 'Can irrational values result in an integer?' ) I tried rearranging the terms and got this: $$[∆ABC]=\boxed{\frac{1}{4}\sqrt{4b^2c^2 -(b^2+c^2-a^2)^2}}$$ This formula is useful because it helps to eliminate the square roots. But, is the original version of Heron's formula more beneficial?
 A: *

*It's not obvious that your formula is symmetric regarding sides. A student may wonder what is so special about side $a$, so one need to subtract it. The original formula is explicitly symmetric, and thus easy to memorize.


*[Edit after Oscar Lanzi comment] Having terms like $(s-a)=\frac12(b+c-a)$ makes it obvious that once triangle inequality fails, the product becomes negative and the formula loses the sense.


*Heron's formula for triangle is a special case of Brahmagupta's formula for area of cyclic quadrilateral, where $d=0$:
$$
A={\sqrt  {(s-a)(s-b)(s-c)(s-d)}}.
$$
If students have learnt Heron's formula, teacher can expand their knowledge with little extra mental efforts.


*Regarding your note

When we have irrational values, solving using conventional method might be too lengthy.

Counterpoint, if we have rational or integer values, then your formula is also not optimal for calculations. With $a=2$, $b=3$, $c=4$, I can do this in my head:
$$
A = \frac14\sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)},
$$
while yours is trickier.
// Btw, according to Wikipedia, your variant of formula is what was discovered by Chinese mathematicians.
