Pythagorean theorem

If only the lengths of the legs of the right triangle are known but not the hypotenuse, then the length of the hypotenuse can be calculated with the equation

If the length of the hypotenuse and of one leg is known, then the length of the other leg can be calculated as

Showing the two congruent triangles of half the area of rectangle BDLK and square BAGF
Right triangle on the hypotenuse dissected into two similar right triangles on the legs, according to Einstein's proof

"If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."

Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:

The following is a list of primitive Pythagorean triples with values less than 100:

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

If instead of Euclidean distance, the square of this value (the squared Euclidean distance, or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates:

Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:

This can be seen from the definitions of the cross product and dot product, as

This can also be used to define the cross product. By rearranging the following equation is obtained

If one erects similar figures (see Euclidean geometry) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.

Pythagoras' theorem in three dimensions relates the diagonal AD to the three sides.

In terms of solid geometry, Pythagoras' theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal BD is found from Pythagoras' theorem as:

This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes.

where the inner products of the cross terms are zero, because of orthogonality.

This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles:

On an infinitesimal level, in three dimensional space, Pythagoras' theorem describes the distance between two infinitesimally separated points as:

The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven