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Rectangles

Rectangle - Diagonals, Formula, Parallelogram, Properties, Perimeter of a rectangle and Area

A rectangle is a two- dimensional regular quadrilateral having vertex angles 90o. What makes a rectangle different from a square are its sides. Unlike a square, a rectangle has opposite sides of equal length which are parallel to each other. The opposite sides of a rectangle are termed as length and breadth or width.

Let ABCD be a rectangle whose length is AB and CD and breadth is AC and BD. Also,

AB = CD = l and AB ∥ CD

AC = BD = b and AC ∥ BD

Where ‘l’ indicates the length and ’b’ indicates breadth or width.

Rectangle as parallelogram

The vertices of adjacent sides of a rectangle always join to make a right angle which indicates that its opposite sides are parallel. This also means that the opposite sides are equal in measure. A quadrilateral with equal and parallel opposite sides is said to be a parallelogram. Hence, a rectangle is also a parallelogram with vertex angles 90°.

Diagonals of a rectangle

A rectangle has two diagonals of equal length. These diagonals bisect each other i.e. divide each other into two equal parts. For example,

Here, in rectangle ABCD, AD and BC are diagonals and O is the point of intersection of the diagonals such that

AD = BC

AO = OD

BO = OC

Length of diagonals: The length of each diagonal can be calculated by a simple formula as below

d = √(l² + b²)

where ‘d’ is the length of diagonal. This formula is derived from the Pythagoras theorem as

we know

(hypotenuse)² = (perpendicular)² + (base)²

In rectangle any rectangle ABCD,

Using Pythagoras theorem,

(AD)² = (AC)² + (CD)²

AD =√[(AC)² + (CD)²]

For AD = d; AC = l: CD = b

d = √(l² + b²)

Properties of a rectangle

  1. It is a 4-sided closed 2-D shape with 4 vertices or corners.
  2. All the vertex angles are equal i.e. 90° each.
  3. Opposite angles are equal which makes it a parallelogram.
  4. Its opposite sides are equal and parallel.
  5. The sum of its interior angles is 360°.
  6. The diagonals of a rectangle are of equal length.
  7. The diagonals bisect each other.
  8. If the length and breadth of a rectangle are equal and diagonals bisect each other at 90° it becomes a square.

Perimeter of a rectangle

The perimeter of a rectangle is the total measure of its boundary. To calculate the perimeter of a rectangle, add all its sides or use the formula

Perimeter of rectangle = 2 . (length + breadth)

Perimeter of rectangle ABCD = 2. (AB + AC)

where length is AB and CD and breadth is AC and BD.

Area of a rectangle

The area of rectangle is defined as the space occupied by a rectangle in a plane. It is calculated as the product of its length and breadth and measured in square units.

Area of rectangle = length . breadth

Area of rectangle ABCD = AB . AC

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