Rectangle
GeometryA 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. Opposite sides are equal in length and parallel. A square is a special kind of rectangle where all four sides are equal.
Example
A door, a book, a dollar bill, and a computer screen are all rectangles. If a rectangle is $6$ cm long and $4$ cm wide, its area is $6 \times 4 = 24$ square cm.
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).
Definition
A rectangle is a parallelogram with four right angles. Properties: opposite sides are congruent and parallel, diagonals are congruent and bisect each other, all four angles $= 90^\circ$. A square is a rectangle with all sides congruent.
Example
Rectangle with length $8$ and width $5$: Area $= 40$, Perimeter $= 26$, Diagonal $= \sqrt{8^2 + 5^2} = \sqrt{89}$ approximately $9.43$. The diagonals bisect each other, so each half-diagonal $= \sqrt{89}/2$.
Key Insight
The diagonal of a rectangle equals $\sqrt{l^2 + w^2}$ by the Pythagorean theorem - the diagonal and two sides form a right triangle. This formula is used in screen size specifications (TV and monitor sizes are given as diagonal measurements).
Definition
A rectangle is a parallelogram with congruent diagonals, or equivalently, a quadrilateral with four right angles. Its symmetry group is the Klein four-group $\mathbb{Z}_2 \times \mathbb{Z}_2$ (two lines of symmetry: horizontal and vertical midlines). The diagonal $d$ satisfies $d^2 = l^2 + w^2$. A rectangle is a cyclic quadrilateral (its vertices lie on a circle with diameter $d$).
Example
The circumscribed circle of a rectangle has radius $R = d/2 = \sqrt{l^2+w^2}/2$, centered at the intersection of the diagonals. For $l=3$, $w=4$: $R = 5/2 = 2.5$. The four vertices of the rectangle lie on this circle.
Key Insight
A rectangle is a cyclic quadrilateral because its opposite angles (both $90^\circ$) are supplementary. Thales' theorem guarantees that any angle inscribed in the semicircle on diameter $d$ is $90^\circ$, explaining why all four corners of a rectangle lie on a circle of diameter equal to its diagonal.