Symbol for area

Question

Is there an accepted convention to denote "area" in the following manner:

$$ A = \frac{1}{3}(\triangle ABC) $$

which would mean, say, "A equals to one third of the area of the triangle ABC? (Or does the above would mean that?)

Answer 1

• $S_{ABC}$. I see this notation fairly often here on Math.SE, but I dislike how it relegates the most important part of the symbol to a subscript. I'm not entirely sure why the "$S$" is used; for "surface", perhaps? Art of Problem Solving's "Area" page suggests subscripted "$A$" or "$K$". (It even mentions that $K_{ABCDEF}$ could denote the area of a hexagon.)

• $[ABC]$ or $[\triangle ABC]$. This notation is not uncommon. I also see it here a good deal. Art of Problem Solving mentions this as alternative notation.

• $(ABC)$ or $(\triangle ABC)$. I've seen this, but only very rarely; that's perhaps because parentheses are already quite overloaded in math notation (and text!).

• $|\triangle ABC|$. This is my notation of choice, as it fairly-obviously extends $|\overline{AB}|$ (and even $|-3|$), so that "$|x|$" means "the measure of $x$", whatever $x$ might be (number, segment, region, etc.). MathWorld uses it in their "Area Principle" entry. (The entry also uses square-brackets to represent "signed" ratios of areas.)

• $(\text{area}\; \triangle ABC)$. This verbose form is perfectly acceptable in situations where it's not worth burdening the audience with an obscure symbol for something you won't use very often anyway. (I used this approach in this answer I recently posted.)

• $T$. If you're going to refer to a particular area many times, assign it a symbol.

Ultimately, you can use whatever notation meets your expositional needs. Given the variety of options out there, you best serve your audience by explicitly defining your usage on first appearance: Here we see that $|\triangle ABC| = 25$, where "$|x|$" indicates the area of $x$.

Answer 2

I would say $$A=\frac 13\cdot A(\triangle ABC)$$