A circle is one of the simplest geometry shapes defined by only a radius. This means that any property of a circle will only depend on its radius. True for a sphere as well both volume and surface area. These properties of these shapes will be proportional to a power of the radius. Now I don’t think I need to adress why circles and spheres are special, these symmetries are quite common.
So we know that the circumference is some a×r and the radius is r. If we look at ar/r = a. So measuring the circumference anf dividing by radius gives us a the proportionality constant. This constant does not depend on r its a constant.
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