The following observations are motivated by the facts that the area of a planar figure displayed on a screen can be expressed by a certain number of pixels; and if the figure is drawn by a plotter, then its area can be characterised by the total length of a line which fills it in.
The generalisations of the Pythagorean theorem are of three kinds. Firstly, the squares on the sides of the right triangle are substituted by other geometrically similar planar figures (Euclid's Elements Book VI, Proposition 31 ). Secondly, the assumption of the right angle is omitted (the law of cosines), or both of these generalizations occur simultaneously (Pappus’ area theorem , see also H. W. Eves ). Thirdly, mathematical spaces other than the plane are considered (for example, de Gua-Faulhaber theorem about trirectangular tetrahedra , further generalised by Tinseau , Euclidean n-spaces, Banach spaces , see also ).