Rectangle
A rectangle is a quadrilateral with four right angles and opposite sides that are equal and parallel.
Formula
\text{Area} = \text{length} \times \text{width}; \text{Perimeter} = 2(\text{length} + \text{width})
Definition
A rectangle is a shape with four sides and four right-angle corners, with opposite sides equal in length and parallel; a square is a special rectangle where all four sides are equal. Formally, a rectangle is a parallelogram with four right angles, or equivalently a parallelogram with congruent diagonals; its symmetry group is the Klein four-group $\mathbb{Z}_2 \times \mathbb{Z}_2$ (two lines of symmetry, horizontal and vertical), and a rectangle is a cyclic quadrilateral, its vertices lie on a circle whose diameter is its diagonal $d$, where $d^2=l^2+w^2$.
Example
A door, a book, a dollar bill, and a computer screen are all rectangles; a rectangle $6$ cm by $4$ cm has area $24$ square cm. For length $8$ and width $5$: area $=40$, perimeter $=26$, diagonal $=\sqrt{8^2+5^2}=\sqrt{89}\approx9.43$, with diagonals bisecting each other so each half is $\sqrt{89}/2$. Its circumscribed circle has radius $R=d/2=\sqrt{l^2+w^2}/2$; for $l=3$, $w=4$: $R=5/2=2.5$.
Key Insight
Rectangles are everywhere in buildings and design because right-angle corners are easy to construct and create stable, efficient structures; the word "rectangle" comes from Latin "rectus" (right) + "angulus" (angle). The diagonal formula $\sqrt{l^2+w^2}$ comes directly from the Pythagorean theorem, which is why TV and monitor sizes are given as diagonal measurements. A rectangle is a cyclic quadrilateral precisely because its opposite angles (both $90^\circ$) are supplementary, and Thales' theorem guarantees any angle inscribed in the semicircle on diameter $d$ is $90^\circ$, explaining why all four corners lie on a circle of that diameter.