Area-Perimeter Relation: Euclidean Objects

Suppose the rectangle has height H and width W. Then

P = 2⋅H + 2⋅W and A = H⋅W

How can we write perimeter ans a function of area? We're faced with two variables, H and W. For which do we solve?

Recall the Area-Perimeter relationship includes a factor, k, that depends on the shape but not the size. The shape of rectangles is determined by the ratio W/H.

So,

A = H⋅W = H2⋅(W/H)

and

H = (√(H/W))⋅A1/2

Next note

P = 2⋅H + 2⋅W = 2⋅(1 + (W/H))⋅H

Combining these, we see

P = 2⋅(1 + (W/H))⋅(√(H/W))⋅A1/2 = k⋅Ad/2

where d = 1 is the dimension of the perimeter, and k = 2⋅(1 + (W/H))⋅(√(H/W)) is a constant that depends only on the snape of the rectangle.

Return to Euclidean objects.