Rectangle
A rectangle is a two-dimensional, closed, four-sided figure with four right angles.[1] It is a type of quadrilateral in which opposite sides are equal in length and parallel to each other, making it a special case of a parallelogram.[2] Key properties of a rectangle include the fact that its diagonals are congruent in length and bisect each other at their midpoints.[3] Additionally, all angles measure exactly 90 degrees, and the figure exhibits bilateral symmetry along the lines connecting midpoints of opposite sides.[4][5] These characteristics distinguish rectangles from other quadrilaterals, such as rhombuses or trapezoids, though a square represents a rectangle where all four sides are of equal length.[6] In practical applications, rectangles are fundamental in geometry for calculating measurements, with the area given by the formula area = length × width and the perimeter by perimeter = 2(length + width).[7][8] These formulas underpin computations in fields like architecture, engineering, and computer graphics, where rectangles model structures, screens, and layouts.[9]Definition and Characterizations
Formal Definition
A rectangle is defined as a quadrilateral in the Euclidean plane with four right angles, each measuring 90 degrees.[10] This configuration ensures that the figure is a closed planar shape bounded by four line segments.[11] This definition is equivalent to that of a quadrilateral possessing four equal interior angles and pairs of opposite sides of equal length, as the right angles imply congruence among all angles and parallelism of opposite sides.[10][12] The term "rectangle" derives from the Latin rectangulus, combining rectus (meaning "right" or "straight") and angulus (meaning "angle"), thus signifying a "right-angled" figure; it entered English usage in the 1570s via French.[13] In ancient geometry, Euclid implicitly referenced rectangles in his Elements (circa 300 BCE) through the concept of a "rectangular parallelogram," described as a parallelogram contained by two straight lines forming a right angle.[14][15] Visually, a rectangle consists of two pairs of parallel sides, with all four angles congruent at 90 degrees, distinguishing it as a specific type of parallelogram.[10]Equivalent Characterizations
A rectangle can be characterized as an equiangular quadrilateral, meaning all four interior angles measure 90 degrees. This equivalence holds because in Euclidean geometry, a quadrilateral with four equal angles must have each angle as a right angle, given that the sum of interior angles in any quadrilateral is 360 degrees.[16] Equivalently, a rectangle is a parallelogram that contains at least one right angle. In a parallelogram, opposite angles are equal and consecutive angles are supplementary; thus, if one angle is 90 degrees, all angles must be 90 degrees. Another equivalent condition is that a parallelogram has congruent diagonals. To see this, consider a parallelogram ABCD with position vectors: let vector \overrightarrow{AB} = \mathbf{u} and \overrightarrow{AD} = \mathbf{v}. The diagonals are \mathbf{u} + \mathbf{v} and \mathbf{u} - \mathbf{v}. Setting their magnitudes equal gives |\mathbf{u} + \mathbf{v}|^2 = |\mathbf{u} - \mathbf{v}|^2, which simplifies to $2\mathbf{u} \cdot \mathbf{v} = -2\mathbf{u} \cdot \mathbf{v}, implying \mathbf{u} \cdot \mathbf{v} = 0. Thus, adjacent sides are perpendicular, confirming it is a rectangle.[17] In vector terms, a rectangle is defined by two adjacent sides represented by vectors \mathbf{u} and \mathbf{v} such that opposite sides are equal and parallel (\overrightarrow{DC} = \mathbf{u}, \overrightarrow{BC} = \mathbf{v}), and adjacent sides are perpendicular (\mathbf{u} \cdot \mathbf{v} = 0). This ensures the figure has the required right angles and parallel sides.[18] In coordinate geometry, an axis-aligned rectangle has vertices at (x_1, y_1), (x_2, y_1), (x_2, y_2), and (x_1, y_2), where x_1 < x_2 and y_1 < y_2. The horizontal and vertical sides are parallel to the axes, and adjacent sides are perpendicular by construction.[19]Classification
Within Quadrilaterals
A rectangle is a special type of parallelogram, characterized by having opposite sides that are both parallel and equal in length, with the additional property of all interior angles measuring 90 degrees.[20] This distinguishes it from a general parallelogram, where angles may not be right angles.[21] Within the category of rectangles, subtypes exist based on side lengths: a square is a rectangle where all four sides are equal, while an oblong, also known as a non-square rectangle, has unequal adjacent sides but retains the four right angles.[22] The square represents the most symmetric subtype, overlapping with other quadrilateral forms.[21] Rectangles differ from other quadrilaterals such as trapezoids, which feature exactly one pair of parallel sides, and kites, which have two pairs of adjacent sides that are equal in length.[20] Unlike these, rectangles require both pairs of opposite sides to be parallel. In the taxonomy of quadrilaterals, rectangles form a subset of parallelograms, which in turn are a subset of all quadrilaterals; rhombuses, with all sides equal, also subset parallelograms, and the square lies at the intersection of rectangles and rhombuses.[20] This hierarchical structure illustrates how rectangles occupy a specific position among convex quadrilaterals with parallel opposite sides.In Broader Geometric Hierarchies
In non-Euclidean geometries, the rectangle's classification extends beyond the Euclidean plane, revealing how its defining properties—such as right angles and opposite equal sides—adapt or fail under different metrics and axioms. In hyperbolic geometry, quadrilaterals with four right angles are impossible because the sum of interior angles in any hyperbolic quadrilateral is less than 360 degrees, a consequence of the constant negative curvature.[23] However, analogous figures known as hyperbolic rectangles can be constructed as equiangular quadrilaterals with four equal angles each less than 90 degrees and opposite sides of equal length; in these, the diagonals are unequal, contrasting the equal diagonals of Euclidean rectangles.[24] In taxicab geometry, also known as Manhattan geometry and governed by the L1 norm where distance is the sum of absolute differences in coordinates, rectangles are defined as four-sided figures with four right angles (measured 90 degrees in the Euclidean sense but interpreted via the taxicab metric) and opposite sides congruent. These "rectangles" exhibit distinct properties from their Euclidean counterparts, such as diagonals whose taxicab length equals the sum of adjacent side lengths rather than the Euclidean hypotenuse, leading to altered notions of symmetry and congruence criteria like SASAS for triangles.[25][26] Beyond plane geometries, the rectangle fits into abstract hierarchies as a special case of higher-dimensional polytopes. Specifically, a rectangle is a two-dimensional orthotope, the generalization of a rectangle to n dimensions defined as the Cartesian product of n closed intervals along mutually orthogonal axes; for example, in three dimensions, this yields a rectangular prism (or cuboid) with all right angles and pairwise perpendicular faces.[27] Rectangles also belong to the family of zonotopes, centrally symmetric polytopes formed as the Minkowski sum of line segments; a rectangle arises from the sum of two perpendicular line segments of appropriate lengths, placing it within broader polyhedral classifications including parallelograms and higher-dimensional analogs like zonoids.[28] These classifications highlight the rectangle's role in unifying geometric structures across metrics and dimensions, with extensions to tessellations in non-Euclidean spaces noted in specialized tilings.[23]Core Properties
Sides and Angles
A rectangle is defined by its four interior angles, each measuring exactly 90 degrees, resulting in a total sum of 360 degrees, consistent with the interior angle sum of any quadrilateral.[6] These right angles ensure that the figure's corners form perfect perpendicular intersections, distinguishing the rectangle from other parallelograms where angles may vary.[29] The sides of a rectangle consist of two pairs of opposite sides that are equal in length, typically denoted as length l and width w, where adjacent sides may differ unless the rectangle is a square.[30] This equality of opposite sides, combined with the property that opposite sides are parallel to each other, positions the rectangle as a special type of parallelogram.[29] The parallelism arises from the right angles, which align the sides such that no side intersects another except at the vertices.[6] Regarding the possibility of an inscribed circle tangent to all four sides, a rectangle admits such an incircle only if it is a square, as non-square rectangles fail the condition that the sums of the lengths of opposite sides must be equal—a requirement established by Pitot's theorem for tangential quadrilaterals.[31] In a general rectangle with l \neq w, the sums are $2l and $2w, which are unequal, preventing the existence of an incircle.[32] This property highlights the rectangle's geometric constraints, briefly noting its duality with the rhombus in quadrilateral classifications.[30]Diagonals and Symmetry
In a rectangle, the two diagonals are congruent in length and bisect each other at their midpoint.[30][33] This bisection occurs because a rectangle is a special type of parallelogram, where the diagonals always intersect at their midpoints, and the congruence follows from the equal right angles at the vertices.[30][33] The rectangle exhibits reflection symmetry across two lines: the horizontal axis passing through the midpoints of the top and bottom sides, and the vertical axis passing through the midpoints of the left and right sides.[34] These reflection symmetries, combined with the identity transformation and a 180° rotation about the center, form the full symmetry group of a non-square rectangle, known as the Klein four-group or dihedral group D_2.[35] Each non-identity element of this group has order 2, reflecting the pairwise commuting nature of the rotations and reflections. In the special case of a square, the symmetry group expands to the dihedral group D_4 of order 8, incorporating additional 90° and 270° rotations as well as reflections over the diagonals.[35] Applying Varignon's theorem to a rectangle, the quadrilateral formed by connecting the midpoints of its sides is a rhombus. This result stems from the theorem's general statement that such midpoints always form a parallelogram, with the additional property of equal side lengths arising specifically when the original quadrilateral has congruent diagonals, as in the rectangle. The rhombus's sides are parallel to the diagonals of the original rectangle and half their lengths, highlighting the rectangle's inherent symmetry in midpoint constructions.Rectangle-Rhombus Duality
In geometry, rectangles and rhombi exhibit a fundamental duality, where the rectangle emphasizes equality among its angles—all four being right angles—while allowing for unequal adjacent sides, and the rhombus prioritizes equality among all four sides while permitting unequal adjacent angles.[36] This complementary relationship trades uniformity in one attribute (angles versus sides) for flexibility in the other, reflecting a broader duality in Euclidean plane geometry between angular and linear measures.[36] The square serves as the fixed point of this duality, satisfying both conditions simultaneously: all angles are right angles, and all sides are equal.[36] In this configuration, the trade-off resolves, yielding a shape with maximal symmetry that aligns with both the rectangle's angular precision and the rhombus's side congruence. This intersection underscores the square's unique position within the family of parallelograms. This duality manifests in optimization problems, such as the isoperimetric inequality for quadrilaterals, where for a fixed area, the square achieves the minimal perimeter among both rectangles and rhombi.[37][38] Specifically, the inequality P^2 \geq 16A holds for any quadrilateral with perimeter P and area A, with equality only for the square, highlighting how deviating from squareness in either direction—increases angular variance in rhombi or side disparity in rectangles—results in larger perimeters for the same area.[37]| Property | Rectangle | Rhombus |
|---|---|---|
| Angles | All equal (90°) | Opposite equal, adjacent supplementary |
| Sides | Opposite equal, adjacent may differ | All equal |
| Diagonals | Equal in length | Perpendicular |
| Symmetry Axes | Bisect opposite sides | Bisect opposite angles |