Rectangle
Definition and Basic Properties
A rectangle is a quadrilateral with four right angles. It is a type of polygon and a parallelogram, as it has two pairs of parallel sides. The opposite sides of a rectangle are equal in length. The word 'rectangle' comes from the Latin 'rectangulus', which is a combination of 'rectus' (meaning 'right') and 'angulus' (meaning 'angle').
Mathematical Properties
A rectangle has several mathematical properties that distinguish it from other quadrilaterals. These properties are derived from the rectangle's defining characteristic: its four right angles.
Area
The area of a rectangle is calculated by multiplying the lengths of its two adjacent sides. This formula is often written as A = l*w, where 'A' represents the area, 'l' the length, and 'w' the width of the rectangle.
Perimeter
The perimeter of a rectangle is calculated by adding the lengths of all its sides. This formula is often written as P = 2l + 2w, where 'P' represents the perimeter, 'l' the length, and 'w' the width of the rectangle.
Diagonals
A rectangle has two diagonals, each of which cuts the rectangle into two congruent right triangles. The diagonals of a rectangle are equal in length and bisect each other. The length of a diagonal can be calculated using the Pythagorean theorem, which states that the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the lengths of the other two sides.
Rectangles in Geometry
Rectangles are fundamental shapes in Euclidean geometry. They are used in the definition and study of many other geometric figures and concepts, including squares, circles, and the Cartesian coordinate system.
Squares
A square is a special type of rectangle where all four sides are equal in length. Because of this, all properties of rectangles also apply to squares.
Circles
In the study of circles, rectangles are often used to define and calculate the properties of inscribed and circumscribed circles. An inscribed circle is a circle that fits inside a rectangle touching all four sides, while a circumscribed circle is a circle that encloses a rectangle touching all four vertices.
Cartesian Coordinate System
In the Cartesian coordinate system, rectangles are used to define the coordinates of points in two-dimensional space. The sides of the rectangle correspond to the x and y axes, and the vertices of the rectangle correspond to the coordinates of points.
Rectangles in Other Fields
Beyond geometry, rectangles have applications in many other fields, including art, architecture, computer science, and physics.
Art and Architecture
In art and architecture, the rectangle is one of the most commonly used shapes due to its simplicity and the sense of stability it provides. The Golden rectangle, a rectangle whose side lengths are in the golden ratio, is considered particularly aesthetically pleasing and has been used in many notable buildings and artworks.
Computer Science
In computer science, rectangles are used in graphical user interfaces to define the shapes of windows, buttons, and other elements. They are also used in algorithms for collision detection, image processing, and other tasks.
Physics
In physics, rectangles are used in the study of motion and forces. For example, a rectangle can represent the path of an object moving at a constant velocity, with one side representing time and the other side representing distance.