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 |
Return to Euclidean objects.