rectangle

A rectangle is a parallelogram with four right angles. In a rectangle, opposite sides are parallel and equal in length, and adjacent sides are perpendicular.

The diagonals of a rectangle are equal in length and bisect each other, dividing the figure into two pairs of congruent triangles.

Area and perimeter: The area is the product of its length and width, A = l × w. The perimeter is P = 2(l + w). If the rectangle is axis-aligned with corners at (x1, y1) and (x2, y2), its area can also be expressed as |x2 − x1| × |y2 − y1|.

Properties: All four interior angles are right angles. A rectangle is a cyclic quadrilateral with all four vertices lying on a common circle, since opposite angles sum to 180 degrees.

Relation to other shapes: A square is a rectangle with all sides equal. A rectangle is a special case of a parallelogram. In vector form, its area can be computed as the magnitude of the cross product of two adjacent side vectors.

Applications: Rectangles are common in architecture, graphic design, tiling, and computer interfaces. They provide convenient, orthogonal shapes for measuring area and developing layouts, coordinates, and geometric reasoning in both theoretical and applied contexts.