Square (algebra)
In geometry

There are several major uses of the square function in geometry.

The name of the square function shows its importance in the definition of the area: it comes from the fact that the area of a square with sides of length l is equal to l. The area depends quadratically on the size: the area of a shape n times larger is n times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a sphere is proportional to the square of its radius, a fact that is manifested physically by the inverse-square law describing how the strength of physical forces such as gravity varies according to distance.

The square function is related to distance through the Pythagorean theorem and its generalization, the parallelogram law. Euclidean distance is not a smooth function: the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance (denoted d or r), which has a paraboloid as its graph, is a smooth and analytic function.

The dot product of a Euclidean vector with itself is equal to the square of its length: v ⋅ v = v. This is further generalised to quadratic forms in linear spaces via the inner product. The inertia tensor in mechanics is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size (length).

There are infinitely many Pythagorean triples, sets of three positive integers such that the sum of the squares of the first two equals the square of the third. Each of these triples gives the integer sides of a right triangle.
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