Area of Rectangle

A rectangle can be considered both a quadrilateral and a parallelogram since it contains two pairs of lines, which are parallel to each other. Opposite lines are of equal lengths, and both the pairs are of different lengths. 

A rectangle can also be considered as an equilateral quadrilateral since each side intersects each other at an angle of 90 degrees. 

Whether or not a quadrilateral is a rectangle or not can be decided if any of the following points are true-

  1. A parallelogram is having a minimum of one right angle. 
  2. A parallelogram which has its diagonals of equal lengths.
  3. Triangles ABD and DCA of a parallelogram ABCD are congruent. 
  4.  A quadrilateral that has four right angles. 
  5. An equiangular quadrilateral. 

Properties

Symmetrical Properties-

  1. Any polygon is cyclic if all of its corners lie on a circle. The rectangle is a cyclic polygon. 
  2. A polygon is equiangular if all of its angles are equal. The rectangle is an equiangular polygon because all of its angles measure an equal 90 degrees. 
  3. A polygon is isogonal if all of its angles lie in a symmetrical orbit. A rectangle is an isogonal polygon.
  4. A polygon is rectilinear if its sides meet at right angles. A rectangle is a rectilinear polygon. 
  5. A polygon is a circumcircle. If its center is equidistant from its vertices, a rectangle is a circumcircle polygon.

Theorems

  1. The isoperimetric theorem- It states that a square has the largest perimeter amongst all the rectangles of a given perimeter.

Crossed Rectangle

A crossed rectangle is a crossed quadrilateral in which there are two opposite sides of a rectangle and two diagonals. Its geometry is similar to a butterfly or a bowtie and is sometimes also called an angular eight because any rectangular frame in 3-d when will be twisted will take the form of a butterfly. Hence the name. 

Geometry of Rectangle

Every rectangle consists of four sides, four angles of 90 degrees each, and two diagonals of identical lengths. In the following figure, we can see the following-

  1. Sides– The sides of this rectangle are AB, BC, CD, DA. 
  2. Diagonals – The diagonals of this rectangle are the sides AC and BD.
  3. Angles – The angles A, B, C, D are all right angles i.e., each of them measures 90 degrees. 

Perimeter

The perimeter of a rectangle is the sum of lengths of all the sides. The following formula can calculate the perimeter of any rectangle. 

  Perimeter= 2*(Length of side 1) + 2*(Length of side 2)

We have only taken sides one and two because the other two sides would be equal in length of one of the other two. So we have multiplied each side by 2 to get the same desired result. 

Consider the following example, If the lengths of sides of a rectangle are given to you say A and B. Then the perimeter of such a rectangle would be 

  Perimeter = (2*A)+ (2*B)

        = 2*(A+B) 

Area

The area of a rectangle is defined as the area covered by the rectangle in a two-dimensional plane. The area of the rectangle can be calculated by using the following formula. 

  Area = (Length*Breadth)

For example, consider the length of the given triangle be A and the breadth be B. The area of such a rectangle will be (A*B).

Relation Between Length, Breadth and Diagonal-

In a rectangle, if we are given any two of the above terms, we can always find the third term. Because in a rectangle, length, its adjacent breadth, and one of the diagonals form a right-angled triangle. So using the Pythagoras theorem, we can always find the area of the rectangle using the following formula. 

Area=

Let’s look at some practical examples to have a clear view of the area of a rectangle and its properties. 

Example 1. The length and breadth of a rectangle are 10 cm and 2 cm, respectively. What is its area?

Answer- Given the Length= 10cm and Breadth= 2cm. We know that area=(Length*Breadth). 

So, Area= (10*2)= 20 cm square. 

Example 2. The length of a rectangle is 3 cm, and the diagonal is 5 cm. What is its area?

Answer- As we have seen in the above formula, we will use the Pythagoras theorem to find the breadth and then multiply it with the length. 

Breadth= UnderRoot(25 – 9)

   = 4 cm

Area= Length*Breadth

       = 12 cm square.  

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