# A formula for the area of a rectangle [closed]

This is a question about elementary geometry, so I think it belongs on this site.

Let $d$ be the length of a diagonal in a rectangle, and let $m$ be half the perimeter.

Then a formula for the area of the rectangle is given by $$A=\frac 1 2 (m^2-d^2).$$

I can easily prove the formula using algebraic manipulations, but I feel it should be possible to see directly from the diagram that the size of the rectangle is half the difference between the sizes of those squares.

So my question is:

What is a simple geometric (and visual) proof of the formula?

• In case this is too mathematical, replace my question with "What is a simple geometric proof of the formula that can be used in school"? – Dag Oskar Madsen May 18 '17 at 15:40
• Straighten out $m$, subdivided by the sides of the rectangle, and make a square. This square contains two copies of the rectangle and also squares of the same side lengths of the rectangle. By Pythagoras (which can be proved visually) those two squares have the same area as the squared diagonal of the rectangle. Thus the difference is two copies of the rectangle. Halve that and you are done. – Reinstate Monica May 18 '17 at 17:49
• I'm voting to close this question as off-topic because I'm not sure what the motivation is. What school audience would seek a purely visual proof of this? What student would be able to follow, for instance, the proof given by @Solmonoff's Secret in their comment, but would not be able to follow the simple algebraic proof? – Brendan W. Sullivan May 19 '17 at 17:58
• @brendansullivan07 I think you are right this is not an ideal way to present geometry to students in school. My motivation was to look at familiar things in new ways, and I was hoping it what came out of it could have value as part of the deeper knowledge we want school teachers to have. But the accepted answer shows this is just a version of Pythagoras, so no real new insight was gained. – Dag Oskar Madsen May 19 '17 at 18:23
• For this site, I really like the archetype of question "What is a good way to explain fact X besides the usual explanation," but I'm not sure "What is your favorite proof of fact X besides the usual proof" is a math education question. It's a fine line and I see it both ways, but I've also voted to close. – Chris Cunningham May 23 '17 at 6:21