Quadrilateral
GeometryA 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. The four inside angles always add up to $360^\circ$.
Example
Squares, rectangles, parallelograms, trapezoids, and kites are all quadrilaterals. A playing card, a book cover, and a window frame are all quadrilateral shapes.
Key Insight
Any quadrilateral can be split into two triangles by drawing one diagonal. Since each triangle has $180^\circ$, the quadrilateral has $2 \times 180 = 360^\circ$ total. This trick works for any polygon.
Definition
A quadrilateral is a polygon with four sides, four vertices, and four interior angles. The sum of interior angles is always $360^\circ$ (since any quadrilateral can be divided into two triangles). Quadrilaterals are classified by their side lengths, parallel sides, and angle properties.
Example
A quadrilateral with angles $80^\circ$, $95^\circ$, $110^\circ$, and $75^\circ$: sum $= 80+95+110+75 = 360$. Hierarchy: square is a special rectangle, rectangle is a special parallelogram, parallelogram is a special trapezoid (in some definitions), all are quadrilaterals.
Key Insight
The quadrilateral family has a rich hierarchy. A Venn diagram of quadrilateral types shows: squares are inside rectangles, which are inside parallelograms, which are inside trapezoids. Understanding this hierarchy helps organize all the special properties.
Definition
A quadrilateral is a simple polygon with $4$ vertices. Its interior angle sum is $(4-2) \times 180 = 360^\circ$. A cyclic quadrilateral (inscribed in a circle) has the additional property that opposite angles are supplementary. The area by shoelace formula is $(1/2)|x_1(y_2-y_4) + x_2(y_3-y_1) + x_3(y_4-y_2) + x_4(y_1-y_3)|$.
Example
Cyclic quadrilateral $ABCD$: angle $A$ + angle $C$ = $180$, angle $B$ + angle $D$ = $180$. This is the converse of the inscribed angle theorem applied twice. Ptolemy's theorem for cyclic quadrilaterals: $AC \times BD = AB \times CD + BC \times AD$ (product of diagonals = sum of products of opposite sides).
Key Insight
Ptolemy's theorem on cyclic quadrilaterals generalizes the Pythagorean theorem (which is the degenerate case when the quadrilateral is a rectangle). The theorem has applications in evaluating trigonometric identities and in proving the triangle inequality for the complex plane.