Why do you square the radius when finding the area of a circle?

Answer 1

Whether you are finding the area of a circle or the area of a square or any other geometrical shape, it is always the case that to go from a linear measurement (a radius, a side, or whatever) to a surface area requires you to use a square of the linear measurement; you are increasing size in two dimensions, length and width, and therefore the effect is the increase the area exponentially, not linearly.