mathschallenge.net logo

Equable Rectangles

Problem

How many rectangles with integral length sides have an area equal in value to the perimeter?


Solution

Let the rectangle measure $x$ by $y$ so that the area = $xy$ and the perimeter = 2x$$ + 2$y$.

therefore $xy$ = 2$x$ + 2$y$
 $$xy$ minus 2$y$$ = 2$x$
 $y$($x$ minus 2) = 2($x$ minus 2) + 4
therefore $y$ = 2 + 4 / ($x$ minus 2)

As $x$ and $y$ are integer, it is necessary for $x$ minus 2 to divide into 4.

Therefore $x$ minus 2 = 1, 2, or 4 implies $x$ = 3, 4, or 6 and $y$ = 6, 4, or 3 respectively.

Hence there are two unique rectangles with area and perimeter equal: an oblong measuring 6 by 3 and a square with side length 4.

Investigate "equable" triangles: equilateral, isosceles, and scalene.

Problem ID: 340 (18 Jun 2008)     Difficulty: 3 Star

Only Show Problem