Square Inches vs Inches Squared

I was just asked the question, is there a difference between 12 square inches and 12 inches squared. At first I assumed that 12 inches squared mean a square with sides of 12 inches. In this case the answer would be no. 12 inches squared would be 144 square inches. I've been searching the web for quite a long time for what is meant by 12 inches squared. I can't find anything. Is there some formal use for the phrase "inches squared"?

Solution 1:

In a context in which you actually expect a square -- such as manufacturing -- one can naturally interpret "12 inches square[d]" as "a square 12 inches on a side". This would be the area of $144$ square inches.

In other context, "inches squared" may be used synonymously with "square inches". This may also depend on the speaker's native language.

Based on comments by Arkamis

Solution 2:

Dr. Math from the Math Forum gives a pretty good explanation here:

It's written as $12 m^2$ (that is, with an exponent of 2), but is better read as "12 square meters" than as "12 meters squared."

The reason for this is as you say, that the latter sounds as if it means $(12 m)^2$, the square of 12 meters, rather than $12 (m^2)$, as it really is, 12 times one meter squared.

The fact is, however, that you will find both forms used; I think it is generally agreed that "12 square meters" is better, in order to avoid that problem, but both are "correct." That is, the problem is not that "12 meters squared" MEANS the area of a 12-meter square, but that it can be taken either way, and is thus ambiguous. We simply want to avoid the ambiguity.

Solution 3:

This topic represents an interesting overlap of mathematics and grammar.

In order to remove ambiguity, if you want “12 inches squared” to equate to the area of a 12” x 12” square, “12 inches” becomes a compound adjective and needs to be hyphenated. Hyphens link words together to remove ambiguity, performing the same function as parentheses in an equation.

If “12 inches squared” refers to a 4” x 3” rectangle, then you would need to hyphenate inches squared, linking those words together to form a compound noun.

However, nobody uses hyphens when writing these mathematical terms, so the only logical way to remove ambiguity is to agree that “12 inches squared” is a square with 12-inch sides, while “12 square inches” is an area that equals twelve 1” x 1” squares. That’s how I have always used these terms.

Mathematics is no place for ambiguity, so a convention needs to be agreed upon. Instead of using inches squared and square inches interchangeably, which obviously creates confusion, area should be measured in square inches (or other units) and abbreviated “sq. in.”