Here are three variations of geometry problems I got from @jamestanton on Twitter.

- The numerical measure of a rectangle’s area and perimeter are equal (P=A) (obviously the units are different). If the rectangle’s sides have integer lengths, what are the dimensions of the rectangle?
- The numerical measure of a box’s surface area and volume are equal (V=SA). If the box’s sides have integer lengths, what are its dimensions?
- The numerical measure of a right triangle’s area and perimeter are equal (P=A). If the triangle’s sides have integer lengths, what are its dimensions?

To give a *complete *solution to a math problem, remember that you must

- show that your proffered solution(s) is (are) correct, and
- shows that no other solutions exist.

Find convincing arguments that you have found * all* of the solutions for each. While my solutions are shown below, I eagerly welcome suggestions for any other approaches.

**STOP!!!
SOLUTION ALERT!!
DO NOT READ ANY FURTHER IF YOU WANT TO SOLVE THESE PROBLEMS ON YOUR OWN.**

**Problem 1:** for a rectangle

Let the *a*=length and *b*=width. Without loss of generality, assume . Then, which implies .

If . If . Thus, the only rectangles for which are a 3×6 and a square with side 4.

**Problem 2:** for a box

Let the *a*=length, *b*=width, and *c=*height. Without loss of generality, assume . Then, which implies .

It wasn’t worth it to find all these by hand, so I wrote a quick spreadsheet to find all *c* values for given *a* and *b* values under the condition .

Therefore, there are 8 such boxes with integer dimensions: 3x7x42, 3x8x24, 3x9x18, 3x10x15, 4x5x20, 4x6x12, 5x5x10, and 6x6x6.

**Problem 3:** for a right triangle

Let the legs be *a* and *b *and the hypotenuse be *c*. Perhaps there is a way to employ the technique I used on the first two problems, but my first successful solution invoked a variation on Euclid’s formula: For some integer values of *k*, *m,* and *n *with , and will form all Pythagorean triples (although not uniquely).

Because , Euclid’s formula gives

What started out feeling like a difficult search for unknown side lengths has been dramatically simplified. can only happen if

- which Euclid’s formula converts to .
- which Euclid’s formula converts to .
- , but this leads to a repeat of the first solution.

Therefore, there are only two right triangles with the property : the 6-8-10 and the 5-12-13 right triangles.

Again, any other solution approaches are encouraged and will be posted.