THE SURFACE OF THE SPHERE Geometry as it is meant to be Imagine a picture completely assembled from small segments, little pieces of straight lines. When you enlarge the picture, every segment will be magnified with the same factor. All enlarged segments have the same ratio relative to each other as before. Imagine another picture assembled from straight and curved segments. Tailors and seamstresses use tape measures. Lay such a tape measure along every segment and make a photograph of the picture. When the photograph is magnified, the number of scale indents along every segment has remained the same. Approach each curved segment by laying a chain of n small straight segments along the curve, n = an integer. The picture now consists of straight segments only. For n approaching to infinity the chain approaches the curve. When you enlarge the picture every segment will be magnified with the same factor, no matter how large n grows. All enlarged segments have the same ratio relative to each other as before.  The circumference of the circle The circumference of a circle is a curve. The diameter of a circle is a segment. When you enlarge or reduce a circle, the ratio of the circumference and the diameter remains the same. Measurements reveal this ratio is 3.1415... 3.1415... is called Circumference / diameter = The diameter of the circle = 2 r This result is not derived, in fact. It is a theorem, an observation. There are a lot of formulas to calculate .