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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.

enter image description here

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?

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closed as off-topic by Tommi Brander, Brendan W. Sullivan, JoeTaxpayer, Chris Cunningham, celeriko May 23 '17 at 13:43

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is off-topic because it is a mathematical question as contrasted with a question about mathematics education. For a Stack Exchange site for mathematical questions please see Mathematics." – Tommi Brander, Chris Cunningham, celeriko
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ 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"? $\endgroup$ – Dag Oskar Madsen May 18 '17 at 15:40
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    $\begingroup$ 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. $\endgroup$ – Solomonoff's Secret May 18 '17 at 17:49
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    $\begingroup$ 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? $\endgroup$ – Brendan W. Sullivan May 19 '17 at 17:58
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    $\begingroup$ @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. $\endgroup$ – Dag Oskar Madsen May 19 '17 at 18:23
  • $\begingroup$ 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. $\endgroup$ – Chris Cunningham May 23 '17 at 6:21
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First copy your rectangle like this to make a big square of side-length m with a square of side-length d drawn inside it.

four rectangles to make some squares

The big square minus the small square leaves four half-rectangles (coloured pink in this picture).

four half rectangles

Half of the pink area is two half-rectangles, and so is the area of one whole rectangle.

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Let m = a + b. Then d^2 = a^2 + b^2. Remove that from m^2 and you have twice the original rectangle. $                                  $

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Here's what I like about this problem, as a math educator. It brings together lots of different aspects of math.

I know that two pieces of information (length and width) determine the size of a rectangle, giving both its area and perimeter. If I know two independent pieces of information about the rectangle, I should be able to find the rest. This related to systems of equations (beginning algebra) and linear algebra.

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