The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. An example calculates the length of the hypotenuse of a triangle with sides of lengths 10 and 6 units. The distance formula calculates the distance between two points in the xy-plane by taking the square root of the sum of the squared differences of the x- and y-coordinates, as shown in an example finding the distance between points (1,4) and (10,16).