Perimeter, Area and Volume

How long and wide must a rectangle be, for its area to equal its perimeter? Assume the width and length are whole numbers.

This is a nice little puzzle to let younger kids explore – if they find the puzzle as interesting or frustrating as I expect, they’ll remember the formulae for perimeter and area for the rest of their lives. For older kids, a bit of algebra gives the answer quickly.

If the length is L and the width is W, and the perimeter equals the area, then LW = 2L+2W, so L = 2W / (W – 2). Pick a width, and this formula gives you the length. Trying a few different values of W, you’ll quickly find there’s only two rectangles with whole number sides, and area equal to perimeter – and one of them is a square.

Well, if the kids are bored of this puzzle, you can try them on a second one – how long, wide and deep must a rectangular box be, for the volume to equal the surface area? Assume the length, width and depth are all whole numbers.

Again, this can be done by trial and error, trying different rectangular boxes, and seeing what their surface area and volume are. Or, those with a bit of algebra can equate the formulae for surface area and volume, and try to find the dimensions of the box. It remains a more interesting puzzle, in ways, for two reasons

  • There are many more solutions, for a start.
  • The algebra is more complicated

For both these reasons, the puzzle is more open-ended – and that’s something that makes puzzles interesting.

In case you need them, the formulae for the surface area and volume of a rectangular box are

  • V = WxDxL (but you knew that, right?)
  • A = 2x(WxD + DxL + LxW)