The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry.

Home Analytic Geometry Triangle Pythagorean Theorem
See also: Properties of a Right Triangle, Generalized Pythagorean Theorem, Thales' Theorem 
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Pythagorean Theorem

The Pythagorean theorem states that for a right triangle the sum of the squared lengths of the shorter sides (legs a and b) is equal to the square of the longest sides (hypothenuse c):

c2 = a2 + b2

Please note that the Pythagorean theorem is invertible: a triangle is a right triangle if the sum of the squared lengths of the two shorter sides is equal to the square of the longest side.

Further, it can be easily shown that there is an equivalent formulation of the Pythagorean theorem which is related to the two non-right angles:

sin2α + cos2α = 1

and

sin2β + cos2β = 1


 

A relation which can be deduced from the Pythagorean theorem is the relation between the squared leg b and the product of the hypotenuse c times the projection of b onto c, p:

b2 = pc

The same is true for a and q (the projection of a onto c):

a2 = qc

Further, it can be shown that

h2 = pq

is true.

Home Analytic Geometry Triangle Pythagorean Theorem

Last Update: 2011-01-11