Tetragons

Tetragons, also known as quadrilaterals, are planar geometric figures bounded by four straight sides and four interior angles. In Euclidean geometry the sum of these interior angles always equals 360°, a property that distinguishes tetragons from polygons with a different number of sides. The term tetragon derives from the Greek words “tetra,” meaning four, and “gon,” meaning angle.

Classification of tetragons depends on side length, angle measure, and symmetry. Common subclasses include squares (four equal sides and right angles), rectangles (opposite sides equal and right angles), rhombuses (all sides equal but angles not necessarily right), and general convex quadrilaterals where no sides or angles are equal. Concave tetragons have one interior angle greater than 180°, while self‑intersecting forms such as crossed quadrilaterals (also called bow‑ties) are non‑simple.

Key theorems concerning tetragons include the Varignon theorem, which states that connecting the midpoints of a quadrilateral’s sides yields a parallelogram, and Brahmagupta’s formula for the area of cyclic quadrilaterals, those that can be inscribed in a circle. A quadrilateral is cyclic if and only if opposite angles sum to 180°. The existence of a unique incircle (tangential quadrilateral) requires the sums of lengths of opposite sides to be equal.

Historically, studies of tetragons appear in Euclid’s Elements and later in the work of Arab mathematicians such as al‑Kashi. Modern applications range from computer graphics, where quadrilateral meshes approximate surfaces, to architectural design, where the structural properties of quadrilateral panels inform material selection.

In non‑Euclidean geometries the angle sum of a tetragon may differ from 360°, reflecting the curvature of the underlying space. Nonetheless, the term “tetragon” remains a fundamental concept across mathematical disciplines, providing a basis for exploring more complex polygonal and polyhedral structures.