Properties of Rectangle

Dhristi JEE 2022-24

What is Rectangle?

A rectangle is a four-sided two-dimensional plane figure. A rectangle is a four-sided polygon with opposing sides that are parallel and equal in length. It is one of the types of quadrilaterals in which all four angles are right angles or equal to 90 degrees. A rectangle is a type of parallelogram in which all of the angles are equal. A square is defined as a rectangle having four equal sides. Now let us learn what are the properties of a rectangle in this article. It is very important for students to understand that the four sides of a rectangle are not equal. A rectangle can never be a square but a square can be a rectangle. We will discuss this in detail later on in this article. Some examples of rectangles we see in our daily life include kites, paintings, and so on. 


Characteristics of a Rectangle

A rectangle is defined by its length (L) and width (W) (W). Both the length and the width are scaled differently. A rectangle ABCD has four sides AB, BC, CD, and DA, as well as right angles A, B, C, and D, as seen in the diagram. The length (L) of the distance between A and B or C and D is defined as the length (L) of the given rectangle, while the width (W) of the given rectangle is defined as the distance between B and C or A and D.


Shape of Rectangle

A rectangle is a two-dimensional flat shape of two dimensions. We can easily represent a rectangle in an XY plane by using the arms of the x-axis and the y-axis to indicate the length and width of the rectangle, respectively.


Types of Rectangles

We are all aware that a rectangle is defined as a geometric figure that has four sides. Out of these 4 sides, the opposite sides are equal to each other and the adjacent sides meet at an angle of 90°. These aforementioned properties are seen in the two types of rectangles  which we will discuss in detail. 


Square

A square is a perfectly symmetrical geometric object. It is a type of rectangle with four equal sides and four equal angles. From the 2D aspect, we can say that the interior angles at each vertex make a total of 90°.  In addition to this, the opposite sides of a square are equal and parallel while the diagonals bisect each other at 90 degrees. Thus, as a rule of thumb, we can say that all squares are rectangles but all rectangles are not squares.


Golden Rectangle 

The golden rectangle is a perfect rectangular figure whose sides are in the golden ratio. This golden ratio refers to a condition where (a + b)/a = a/b, where 'a' is the width of the figure and a +b is the length of the rectangle. It would not be wrong to thus say that a golden rectangle is a rectangle whose length to width ratio is the golden ratio of 1: \[(1+ \sqrt{\frac{5}{2}})\]. The Golden Ratio will always come up to something around 1: 1.618. 


Rectangular Shaped Objects

Televisions, computer screens, notebooks, cell phones, CPUs, notice boards, tables, books, TV screens, mobile phones, walls, magazines, tennis courts, and so on are some of the most popular rectangular everyday items we see.


Properties of Square and Rectangle

Here we have listed the Properties of square and rectangle

Properties of a Square

The following are the most important properties of a square:

  • All four interior angles are equal to 90°

  • The square's four sides are congruent or equal to one another.

  • The square's opposite sides are parallel to each other.

  • The square's diagonals are 90 degrees apart and bisect each other.

  • The square's two diagonals are identical in length.

  • The square has four sides and four vertices.

  • The square's diagonal divides it into two identical isosceles triangles.

  • The length of the diagonals exceeds the length of the square's sides.


Fundamental Properties of a Rectangle

If you want to write the properties of a rectangle, then take a look at the following pointers to gain a better understanding. 

  • A quadrilateral is a rectangle.

  • Each interior angle is 90 degrees and the opposite sides are parallel and equal to each other.

  • The sum of all the interior angles is equal to 360 degrees

  • The diagonals are parallel to each other and have the same length.

  • The perimeter of a rectangle with side lengths a and b is 2a+2b units.

  • The area of a rectangle with side lengths a and b is ab sin 90 = ab square units.

  • The diameter of a rectangle's circumcircle is diagonal.

If a and b are the sides of a rectangle, then the length of each diagonal is: 


\[d = \sqrt{a^{2}+b^{2}}\]


The diagonals bisect each other at different angles. One is an acute angle, and the other is an obtuse angle. The rectangle is known as a square if the two diagonals bisect each other at right angles.


When the rectangle is rotated along the line connecting the midpoints of the longer parallel sides, it forms a cylinder. The height of the cylinder is equal to the width of a rectangle in this situation. In addition, the diameter of a cylinder is equal to the length of a rectangle. When the rectangle is rotated along the line connecting the midpoints of the shorter parallel sides, the result is a cylinder. In this case, the cylinder's height is equal to the rectangle's length. Similarly, the width of a rectangle is equal to the diameter of a cylinder. In the above point, we have discussed all properties of rectangles. Students should know all these properties.


The Perimeter of a Rectangle

The diameter of a rectangle is the cumulative distance protected by the rectangle's outer boundary. It is counted in units of time. 

The perimeter formula is as follows:

Perimeter, P = 2 (Length + Width) 


Area of Rectangle

The region covered by a two-dimensional shape in a plane is called the area. It is based on square units of measurement. As a result, the rectangle's area is equal to the area enclosed by its outer boundaries. It equals the sum of the length and width.

The formula of area of a rectangle is:

A = Length x Width  units2


Length of Diagonals of Rectangle

The rectangle is an asymmetrical shape with equal length diagonals on both sides. The rectangle will be divided into two right-angle triangles by a diagonal. As a result, the Pythagorean Theorem can be used to conveniently measure the length of diagonals, where the diagonals are treated as the right triangle's hypotenuse


Let D be the hypotenuse, and L and W, respectively, be the base and perpendicular. As a result, the rectangle's diagonal length would be 


\[d = \sqrt{L^{2}+W^{2}}\]


Solved Examples

1. Calculate the Circumference of a Rectangle With Sides of 10cm and 5cm.

Solution:

Given: a = 10 cm and 5 cm

According to the properties of a rectangle, we know that the perimeter of a rectangle is: Perimeter, P = 2a +2b units =2(a+b) units

Now, substitute the valuesP=2(10+5)P=2(15)P=30 cm

Hence, the perimeter of a rectangle is 30 cm.


2. Find the Length of the Diagonal of a Rectangle Where Length and Width Are Given as 12 and 9 Cm respectively.

Solution: Given length =12 cm and width = 9cm

We know the diagonal formula 

\[d = \sqrt{L^{2}+W^{2}}\]

\[\Rightarrow d=\sqrt{12^{2}+9^{2}}\]

\[\Rightarrow d=\sqrt{144+81}\]

\[\Rightarrow d=\sqrt{225}\]

\[\Rightarrow d= 15cm\]


Conclusion

In this article, we have learned the definition of a rectangle and its properties. We can conclude that all squares are rectangles, but not all rectangles are squares. Since it has all of the properties of a rectangle, a square is a special type of rectangle. We know what are the properties of a rectangle and also we have solved a few example problems.

FAQs on Properties of Rectangle

1. What is the Shape of a Rectangle?

A rectangle is a 2-dimensional figure. It is a closed flat shape with four sides. The four vertices make angles of 90 degrees on each side. The rectangle's opposite sides are identical and parallel and the shape can easily be represented in the XY plane. This can be done by using the arms of the two axes to indicate the length and width of the rectangle. Although a square is a type of rectangle, a rectangle is a type of parallelogram in which all of the angles are equal.

2. What is the Difference Between a Rectangle and a Square?

Via the means of this article, we have all clearly understood that the rectangle and the square both are shapes that have four sides and four internal angles. They are closed geometrical figures each having angles that round up to 90 degrees. The four sides of a square are all the same length. The adjacent sides of a rectangle, on the other hand, have different lengths. The opposing sides are on a level playing field. You should go through the properties of both these shapes to get a clearer idea about the distinction between these figures. 

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