The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The distance formula calculates the distance between two points by taking the square root of the sum of the squared differences between the x-coordinates and y-coordinates of the two points. Examples are provided to demonstrate applying the Pythagorean theorem to find the length of the hypotenuse of a right triangle and using the distance formula to calculate the distance between two points.