Fourangles

Fourangles is a term sometimes used informally to describe polygons with four interior angles, i.e., quadrilaterals. In Euclidean geometry, any four-sided polygon has four interior angles whose measures sum to 360 degrees.

Fourangles can be convex, where all interior angles are less than 180 degrees, or concave, where one or more angles extend inward. Among fourangles, cyclic quadrilaterals have the additional property that opposite angles sum to 180 degrees. The classification by side and angle relations yields various common types such as parallelograms (opposite sides parallel), rectangles (four right angles), rhombuses (four equal sides), squares (four equal sides and four right angles), trapezoids (at least one pair of parallel sides), and kites (two distinct pairs of adjacent equal sides).

The diagonals of a fourangle reveal many of its features: in a parallelogram they bisect each other; in a rectangle they are equal in length; in a rhombus they are perpendicular. In a square they are both equal and perpendicular. These diagonal properties help distinguish subclasses when side lengths and angle measures are known.

Area can be computed by dividing the shape into triangles, or using the shoelace formula from vertex coordinates. Special cases yield simple formulas, such as area = base × height for a rectangle or area = s^2 for a square. For a general fourangle, the area depends on the arrangement of vertices and can be obtained from coordinate methods or decomposition into simpler shapes.

Although widely understood as a four-sided polygon, the term fourangles is not standard in most mathematical literature; quadrilateral remains the preferred term. See also quadrilateral, polygon, and shoelace formula.