Why are area and perimeter not dependent on one another?

Answer 1

That's because you can easily have two different shapes with the SAME perimeter, and DIFFERENT areas, or vice versa. Here is an example:* A 2x2 rectangle has an area of 4, and a perimeter of 8.

* A 1x3 rectangle has an area of 3, and a perimeter of 8.

* A 0x4 rectangle has an area of 0, and a perimeter of 8. (If you don't like this rectangle, you can make one that is arbitrarily close, i.e., a very small width.)

Note that for two SIMILAR figures, any linear measurements are proportional to the scale size, and any area measure is proportional to the square of the scale size - that will make the area proportional to the perimeter, but only for two similar shapes, e.g., two rectangles with the same length-to-width ratio.

Answer 2

For regular polygons , circles, and some other shapes they are dependent. But in general, the two measure different characteristics which results in the absence of dependence.