# Sum of squares

In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts:

## Algebra, algebraic geometry, and optimization

• Polynomial SOS, polynomials that are sums of squares of other polynomials
• The Brahmagupta–Fibonacci identity, representing the product of sums of two squares of polynomials as another sum of squares
• Hilbert's seventeenth problem on characterizing the polynomials with non-negative values as sums of squares
• Sum-of-squares optimization, nonlinear programming with polynomial SOS constraints
• The sum of squared dimensions of a finite group's pairwise nonequivalent complex representations is equal to cardinality of that group.

## Euclidean geometry and other inner-product spaces

• The Pythagorean theorem says that the square on the hypotenuse of a right triangle is equal in area to the sum of the squares on the legs. The sum of squares is not factorable.
• The squared Euclidean distance between two points, equal to the sum of squares of the differences between their coordinates
• Heron's formula for the area of a triangle can be re-written as using the sums of squares of a triangle's sides (and the sums of the squares of squares)
• The British flag theorem for rectangles equates two sums of two squares
• The parallelogram law equates the sum of the squares of the four sides to the sum of the squares of the diagonals
• Descartes' theorem for four kissing circles involves sums of squares
• The sum of the squares of the edges of a rectangular cuboid equals the square of any space diagonal