Pythagorean Theorem

Square the two legs, add the squares and take the square root: c = √(a² + b²). The hypotenuse c is always the longest side and lies opposite the right angle. Example: for legs of 5 and 12, c = √(25 + 144) = √169 = 13. All lengths must be in the same unit. Make sure to square and add first and only then take the root; the root of a sum is not the sum of the roots. The theorem holds only in right triangles, so you first identify the right angle and the side opposite it.

Rearrange the theorem for the required leg: a = √(c² − b²). So subtract the square of the known leg from the square of the hypotenuse and take the root of the result. The order matters, because here there is a minus, not a plus as for the hypotenuse. Example: for c = 13 and b = 5 it follows that a = √(169 − 25) = √144 = 12. Make sure to put the square of the hypotenuse first; if the expression under the root is negative, you have confused hypotenuse and leg. The hypotenuse is always the longest side, so its square is the largest.

The theorem applies only in right triangles in Euclidean, that is planar geometry. In a triangle without a right angle, a² + b² does not equal c²; there you need the law of cosines instead, which as a generalization contains an extra term with the included angle. On curved surfaces, for example on a sphere, the theorem also does not hold, because there the sum of angles in a triangle differs from 180°. A common mistake is to apply the theorem to arbitrary triangles. Therefore always check first whether there really is a right angle before using a² + b² = c².

The distance between two points in the plane is the hypotenuse of a right triangle whose legs are the differences of the coordinates. For the points (x₁, y₁) and (x₂, y₂), d = √((x₂ − x₁)² + (y₂ − y₁)²). So you form the differences in the x and y directions, square them, add them and take the root. In space the term (z₂ − z₁)² is simply added under the root. This distance formula is a direct application of the Pythagorean theorem and the basis for vector lengths, the magnitude of a displacement and many problems in analytic geometry.

A Pythagorean triple consists of three natural numbers a, b and c that satisfy the equation a² + b² = c² exactly. The best known is 3, 4, 5, since 9 + 16 = 25. Others are 5, 12, 13 and 8, 15, 17. Multiples of a triple, such as 6, 8, 10, are also triples. Such numbers are handy because they give clean, whole-number results in problems and let craftsmen set out a right angle without a protractor, for example with a twelve-knot rope. If you recognize a known triple in a problem, you can read off the third side immediately without having to compute the square root.