Such a proof depends on your definition of how you measure the length of the circumference. One way is to find a series of polygons whose vertices lie on the circle, find the perimeter of the polygons and find the limit that the perimeter approaches. Suppose you have two circles, one diameter 1, one diameter 2. You can repeat the polygon process with both circles. At each stage, the polygon for circle 2 has twice the perimeter of the polygon for circle 1. In the end, the circumference of circle 2 is twice that of circle 1, so the ratio of diameter to circumference is the same for both circles. This procedure applies to all possible pairs of circles, so the ratio is pi for all circles. How do we know the polygon perimeters are in the same ratio as their diameters? Draw the two polygons with the same center and join the center with straight lines to two adjacent vertices of the larger polygon. These lines enclose two triangles, one small one large. Each triangle has two sides which are both equal to the radius of a circle, so in each case the triangle is isoceles The two triangles are similar: they have one angle at the center of the circles in common, and as they are both isoceles the other angles are all the same too. As the triangles are similar, the opposite sides, and therefore the perimeters are also in the same ratio as the diameters.