# Math Help - related to area and perimeter

1. ## related to area and perimeter

Q. Find the sides of all rectangles having integral sides whose area is numerically equal to the perimeter?

2. 4, but why?

3. Well, area = l x w = lw

Perimeter = 2l + 2w

When they are equal, we get:

lw = 2l + 2w

0 = 2l - lw + 2w

-2w = l(2-w)

$l = \dfrac{2w}{w-2}$

$l = 2 + \dfrac{4}{w-2}$

Since $2 + \dfrac{4}{w-2}$ must be an integer, w can only be equal to 3,4 or 6. Any larger value will give rise to a fraction.

Hence, when the width is 3, the length becomes 2+4 = 6

When the width is 4, the length becomes 2+2 = 4

When the width is 6, the length becomes 2+1 = 3

That's it!