Quadrilateral

A quadrilateral is a polygon with four sides and four vertices of a polygon. For a simple quadrilateral, its interior angles sum to $2\pi$ by dividing it along an interior diagonal of a polygon into two triangles. Rectangles, squares and parallelograms are special cases. The qualifier convex quadrilateral excludes a reflex interior angle; a self-crossing four-segment chain requires a different interior convention. Four side lengths alone generally leave freedom to vary the angles and area.

A convex quadrilateral is a simple quadrilateral whose filled region is a convex set. In the nondegenerate case, each interior angle is strictly less than $\pi$, and its two diagonals of a polygon meet inside the region. A rectangle is a convex quadrilateral; a dart-shaped concave quadrilateral is not. If the diagonal lengths are $p,q$ and the angle between them is $\theta$, splitting about their intersection gives area $pq\sin \theta /2$. This is at most $pq/2$, with equality for perpendicular diagonals.

A Euclidean polygon with four sides and four right angles. Opposite sides are parallel and have equal lengths. A rectangle is a square exactly when its adjacent side lengths agree.

A rectangle with all four side lengths equal. It is a regular four-sided polygon. Rotations and translations preserve this definition; its sides need not be parallel to the coordinate axes.

A square lamina is a frame obtained by removing a concentric square hole from a larger square, with corresponding sides parallel. If their side lengths are $a>b>0$, the uniform border thickness is $t=(a-b)/2$ and the frame has areaFor positive integer sides $a,b$ with the same integer parity, $t$ is a positive integer, and a unit-grid frame contains exactly $4t(b+t)$ unit squares. For example, outer side five and inner side three give thickness one and sixteen unit squares. Concentricity alone would allow a rotated inner square; the parallel-side condition is what makes this a uniform lattice border.

The unit square here is the Cartesian product $[0,1]\times [0,1]$ in the Euclidean plane, a square of side length one.

A parallelogram is the convex set with vertices $p,p+u,p+v,p+u+v$ for two linearly independent planar vectors $u,v$. Its opposite edges are parallel and its centre $p+(u+v)/2$ gives central symmetry. Its area is $|\det [u,v]|$, so an invertible linear map changes its area by the absolute value of that map's determinant. These facts make parallelograms convenient convex bodies for two-dimensional applications of the Minkowski convex body theorem.

For a quadrilateral with side lengths $a,b,c,d$, semiperimeter $s$, area $K$, and opposite angles $A,C$,The second term vanishes exactly when $A+C=\pi$, which is the cyclicity condition for a nondegenerate convex quadrilateral.