In modern Euclidean geometry, a triangle's area, K, given by Heron's formula, and its sidelengths a, b, and c are considered primary. Therefore, the altitude from a vertex to the opposite has the form:

      hA   = 2K/a

where A is the vertex and a is the sidelength opposite A.

Altitudes are important in many geometric proofs. They are also used constructively to create circles within triangles and quadrilaterals.

A few key facts about altitudes: