Quadrilateral
A quadrilateral is a polygon with four sides, four vertices, and four interior angles summing to 360 degrees.
Formula
\text{angle sum} = 360^\circ
Definition
A quadrilateral is any flat shape with exactly four straight sides and four corners, and its four inside angles always add up to $360^\circ$, since any quadrilateral can be divided into two triangles by a diagonal, each contributing $180^\circ$ (a trick that generalizes to any polygon: interior sum $=(n-2)\times180$). Quadrilaterals are classified by side length, parallel sides, and angle properties, forming a hierarchy: squares are special rectangles, rectangles are special parallelograms, and parallelograms are special trapezoids (in some definitions).
Example
Squares, rectangles, parallelograms, trapezoids, and kites are all quadrilaterals; a playing card, a book cover, or a window frame are quadrilateral shapes. A quadrilateral with angles $80^\circ$, $95^\circ$, $110^\circ$, and $75^\circ$ sums to $360^\circ$. A cyclic quadrilateral (inscribed in a circle) has opposite angles supplementary, and Ptolemy's theorem states $AC \times BD = AB \times CD + BC \times AD$ (product of diagonals equals sum of products of opposite sides).
Key Insight
A Venn diagram of quadrilateral types shows squares nested inside rectangles, nested inside parallelograms, nested inside trapezoids, a hierarchy that helps organize all their special properties. Ptolemy's theorem on cyclic quadrilaterals generalizes the Pythagorean theorem, which is the degenerate case when the quadrilateral is a rectangle, and it has applications in evaluating trigonometric identities and proving the triangle inequality in the complex plane.