What is RECTANGLE?
A rectangle is a quadrilateral with equal opposite sides and four right angles, as defined by Euclidean Plane Geometry. In other words, a rectangle is a parallelogram with a right angle (90°) or an equiangular quadrilateral (it is a quadrilateral whose all angles are equal, i.e., 360°/4 = 90°). If the length of the four sides of the rectangle is equal then it is called a square. Sometimes, a rectangle that is not a square is called an oblong. The rectangle given below has four vertices named ABCD. So, it will be denoted as ABCD.
The word rectangle is derived from the Latin word ractangulus. It is made up of two words; rectus and angulus. Rectus means an adjective, right, or proper while angulus means an angle.
There are certain characteristics that are required in a convex quadrilateral to make it a rectangle. It can be possible if and only if, the quadrilateral follow any one of the following characteristics:
- It is made up of four sides with four vertices.
- The angle of each vertex is a right angle, i.e., 90°.
- Its opposing sides are equal and, as a result, parallel.
- A rectangle, in other terms, is a parallelogram with four right angles.
- The triangles ABD and DCA are congruent in this parallelogram ABCD.
- An equilateral quadrilateral is a rectangle.
- This equilateral quadrilateral has two diagonals that always intersect each other at its center point.
- All interior angles of the rectangle have a sum equal to 360°.
- In the convex quadrilateral with an area of successive sides a, b, c, d equal to:
¼ (a + c) (b +d)
½√(a2 + c2) (b2 + d2)
Properties of Rectangle
1. Duality of Rectangle-Rhombus: When the midpoints of the sides of the rectangle are joined, the rhombus is formed. Therefore, the rhombus is the dual polygon of the rectangle. The difference between a rectangle and rhombus is given below:
|It has equal angles.
||It has equal alternate angles.
|It has equal alternate side.
||It has equal sides.
|It has a circumcircle due to the equal distance between its center and vertices.
||It has an incircle due to the equal distance between its center and sides.
|The opposite sides are bisected by two axes of symmetry.
||The opposite angles are bisected by two axes of symmetry.
|The length of the diagonals is equal.
||The angle at which diagonals intersect is equal.
- All corners of the rectangle lie on a single circle. Therefore, it is called cyclic.
- As all corner angles are equal, i.e., right angle, therefore the rectangle is called equiangular.
- All corners of the rectangle are placed within the same symmetry orbit. Therefore, it is called isogonal or vertex-transitive.
- The reflection symmetry and rotational symmetry of order 2 apply to the two lines of the rectangle.
3. Others: Here are some other properties of the rectangle:
- As we know the sides of the rectangle meet at the right angle, hence it is rectilinear.
- Two rectangles are known as incomparable if none of them fit inside the other.
- Traditional Hierarchy: A rectangle is a special case of the parallelogram which has two pairs of opposite sides. In the rectangle, these pairs of opposite sides are perpendicular to each other, hence form right angles. A parallelogram is a special case of trapezium whose two opposite sides are parallel to each other and have equal length. And a trapezium is a special case of a convex quadrilateral which has at least a single set of opposite sides parallel to each other. There are two qualities of a convex quadrilateral:
Hence, this traditional hierarchy can easily be understood from the given figure.
- Start-shaped (as the whole interior can be seen from a single point and there is no need to cross any edge for this), and
- Simple (as its boundary does not cross itself).
- Alternative Hierarchy: In De Villiers' words, any quadrilateral which has access to symmetry via each pair of contrary sides is termed as a rectangle. This definition states both rectangles, i.e., right-angled and crossed. Each of them has a symmetrical parallel axis to and at an equal distance from a pair of opposite sides. Another axis is the perpendicular that bisects those sides. But in crossed rectangle, the conditions of the first axis are not similar because it does not follow a symmetrical pattern to either bisecting side.
Quadrilaterals which have two symmetrical axes, each from a set of opposite sides, relate to the larger class of quadrilaterals with having a minimum of a single symmetrical axis running from a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia.
If the rectangle has a length of l and width of w, the following formulas apply:
- Rectangle Perimeter: The perimeter of a rectangle is the total distance covered by its outside boundary. The unit of the measurement of the perimeter is length, i.e., kilometer, meter, centimeter, etc. It can be measured by multiplying the addition of length and breadth of the rectangle. This can be understood as follows:
Perimeter, P = 2 (Length × Breadth)
- Area of Rectangle: The total region covered by the two-dimensional shapes, i.e., the length and breadth of the rectangle are known as its area. Its dimensions are measured in square units. As a result, the rectangle's area is defined as the area enclosed by its outer perimeter. It is calculated by multiplying the length and width together. The formula of the calculation is as under:
Area, A = Length × Breadth
- Diagonals of Rectangle: The two diagonals of the rectangle always intersect or cross each other. These diagonals are the same length. The rectangle is divided into two right-angle triangles by these two diagonals. Hence, the length of these diagonals can easily be found by using Pythagoras Theorem. In this theorem, the diagonals are used as the hypotenuse of the right triangle to find their length. This can be understood from the given formulae:
D = (L2 + W2)
Here, D refers to the hypotenuse of the triangle, L stands for the length which is the base of the triangle and W is width which is the perpendicular of the triangle.
- A parallelogram whose diagonals have equal length is known as a rectangle.
- Any quadrilateral that has perpendicular diagonals from the midpoints of its sides forms a rectangle.
- As per the isoperimetric theorem, the area of the square is the largest among all the rectangles with the same perimeter.
- For each point P in the same plane of the rectangle with vertices A, B, C, and D, the British Flag Theorem states:
(AP) 2 + (CP) 2 = (BP) 2 + (DP) 2
Types Of Rectangle
- Crossed Rectangles: A crossed quadrilateral is a figure with two opposite sides of a non-self-intersecting quadrilateral that has two diagonals. A crossed rectangle, like a simple rectangle, has two opposite sides that are parallel to each other, as well as two diagonals. The vertex arrangement of the crossed rectangle is similar to the simple rectangle. This rectangle contains two identical triangles that share a common vertex at the meeting point of the diagonals. But as per the geometric intersection, this vertex is not to be considered.
The best example of a crossed rectangle is the bow or butterfly. They are sometimes also termed angular eight. By twisting a 3D rectangular wireframe, the shape of a bow tie can be formed. The two angles of the crossed rectangle are reflexes and two are acute. Despite the fact that the figure is not equiangular, each pair of opposite angles is equal. This crossed quadrilateral's interior angles add up to 720°.
Some of the common properties of a rectangle and a crossed rectangle are as under:
- The length of opposite sides is equal.
- The length of the two diagonals is equal.
- This rectangle contains two lines of rotational symmetry and reflectional symmetry of order 2.
- Spherical Rectangle: As per the spherical geometry, a figure that has four edges with great circle arcs is known as the spherical rectangle. At the point at which these arcs bisect each other, equal angles are formed which are greater than a right angle. The length of opposite arcs is always equal in a spherical rectangle.
- Elliptic Rectangle: As per the elliptic geometry, a figure in the elliptic plane having four edges that are formed with the elliptic arcs is known as the elliptic rectangle. At the point at which these arcs bisect each other, equal angles are formed which are greater than a right angle. The length of opposite arcs is always equal in an elliptic rectangle. The simplest form of elliptic geometry is spherical geometry.
- Hyperbolic Rectangle: As per the hyperbolic geometry, a figure in the hyperbolic plane having four edges that are formed with the hyperbolic arcs is known as the hyperbolic rectangle. The point at which these arcs bisect each other, the equal angles are formed which are less than a right angle. The length of opposite arcs is always equal in hyperbolic rectangle.
Some Objects With Rectangular Shape
There are many things that we see in our day to day life have rectangular shape. Such things include, television, tray, notebook, table, phone, newspaper, cricket pitch, CPU, wall, etc.