Mensuration

Rectangles and Squares: Definition, Properties & Perimeter

Introduction

Every one of us must have played carrom or chess. What is the shape of the carrom board or the chessboard? Is it a square or a rectangle? What about the tiles of the chessboard, and the tiles of the floors in our houses? We are surrounded by rectangular and square shaped objects. In geometry, we study these shapes and many other shapes like triangles, circles, ovals, and so on. Let’s learn more about the definition of rectangle and square, properties of rectangle and square, and formulae like perimeter and area of these two shapes.

Table of Contents

Definition of Rectangle

Properties of Rectangle

Definition of Square

Properties of Square

Rectangular Shaped Objects

Square Shaped Objects

Perimeter of a Rectangle

Perimeter of a Square

Area of a Rectangle

FAQs on Rectangle and Squares

Rectangles

Definition of Rectangle

The definition of rectangle is given as: a plane shape with four sides. It is a 4 sided polygon with opposite sides parallel to each other. (Note: A square is a special rectangle. We’ll learn the definition of square in an upcoming section).

Properties of Rectangle

Now that we have seen the definition of rectangle, let’s understand the important properties of rectangle. Looking at the above figure we see the opposite sides (DA and CB) are parallel and equal in length. If we look at the other side, we see that AB and DC are also parallel to one another and equal in length. So in this figure, the opposite lines are parallel to one another.

The sides DA and CB have the same length, so it is clear that they are congruent. Also, side DC and AB are congruent to one another.

As per definition of rectangle, this figure has four angles. So, one of the properties of rectangle is that all the angles in a rectangle are 90°.

So we can write it as m∠A = m∠B = m∠C = m∠D = 90°.

Another important one of the properties of rectangle is that the adjacent angles are supplementary.

That is, 90° + 90° = 180°. The sum of all the interior angles is 90° + 90°+ 90° + 90° = 360°

The diagonals of the rectangle are also congruent to each other and they bisect each other at their point of intersection.

A rectangle can also be called a quadrilateral as it has 4 sides, but not all quadrilaterals fit the definition of rectangle.

Formula for area of rectangle = length × breadth

We can summarise the 3 basic properties of rectangle as follows. Compare these with properties of square discussed in an upcoming section:

  • All angles are 90°.
  • Opposite sides are parallel and equal in length.
  • Diagonals bisect each other.

Examples of rectangular shaped objects are covered in a section below.

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Squares

Definition of Square

Square is a quadrilateral in which all the sides have equal length and all the four corners are at right angles. (Note: A square also fits into the definition of rectangle).

Properties of Square

Based on the definition of square, we can understand the following properties. Compare these with the properties of rectangle we learnt above.

  • One of the important properties of square is that the opposite sides are parallel, with all sides being equal.
  • A square has four lines of symmetry.
  • The order of rotational symmetry is 4.
  • The diagonals bisect each other at 90° or right angles. This is one of the properties of square that make it different from rectangles.
  • Opposite sides are equal and parallel (this matches the definition of rectangle as well).
  • All angles are equal to 90°.
  • The diagonals are equal.
  • Diagonals bisect the angles (this is also one of the important properties of rectangle)
  • Any two adjacent angles add up to 180°.
  • Each diagonal divides the square into two congruent isosceles right-angled triangles.
  • The sum of the four exterior angles is 4 right angles.
  • The sum of the four interior angles is 4 right angles.

Thus we have covered the definition of square, and its properties. Find examples of square shaped objects in an upcoming section.

Rectangular Shaped Objects

We have seen the definition of rectangle as a shape with four sides, with opposite sides being parallel and equal in length. We also learnt the properties of rectangle. Now, let’s look at examples of everyday rectangular shaped objects: books, laptops, table tops, doors, notice boards, and so on.

  1. Books

If you observe a book from the top or the bottom, you will see that it is shaped in the form of a rectangle. A rectangle is a 2D shape. When considering the overall shape of a book, it is a cuboid (3D), as it has depth. But when we only consider the shape looking from the top or the bottom, it matches the definition of rectangle. So, books are rectangular shaped objects.

 

  1. Laptop

A laptop too fits the definition of rectangle. It has four sides, and opposite sides are equal and parallel to each other. Even the screen of a laptop is rectangular in shape. Laptops are one of the most common rectangular shaped objects we see in daily life.

  1. Table Tops

A table top that has four sides, and opposite sides are equal and parallel, is rectangular in shape. Table tops are common rectangular shaped objects. However, there are tables that are shaped as circles or ovals as well.

  1. Doors

Doors are often rectangular in shape. Even most windows are rectangular shaped objects

 

  1. Notice Board

Another object that matches the definition of rectangle is a notice board, seen every day at school.

Square Shaped Objects

Just like rectangular shaped objects we see around us, there are thousands of square shaped objects too. Here are a few examples of square shaped objects that fit the definition of square:

  1. Tissue Papers

The definition of square is a shape with four equal sides and each angle is a right angle. Paper napkins, commonly called tissue papers, are examples of square shaped objects.

 

  1. Tiles

Tiles are another example of square shaped objects in our daily lives. Floor and wall tiles often come in square shape.

 

  1. Photo Frames

Some photo frames also fit the definition of square.

  1. Chess Board

Chess board and the tiles on a chess board are square shaped objects.

 

These were examples of objects that have the properties of square. But it is important to understand that the shape of these objects matches the definition of rectangle too, as they have four parallel sides, opposite sides are of equal length, and the internal angles are right angles.

Perimeter of a Rectangle

The perimeter of a rectangle refers to the border or boundary of a rectangle. It is one of the most important formulas in mensuration. To understand the meaning of perimeter, consider you are standing at one corner of a rectangle. Earlier, we considered examples of rectangular shaped objects. Imagine any one of these objects. Now, start moving across the line or edge of the rectangle so that you come back to the starting point. The total length that you travelled is the perimeter of a rectangle. Every geometric shape which is 2D will have a perimeter. In simple words, we can say that the perimeter is the length of the border of the shape or figure.

For different shapes, there are different formulae for perimeter. For instance, the perimeter of a circle, also called circumference, is given as 2 π r (where r is the radius). For a square, the perimeter is calculated by adding the length of the four sides or by multiplying the length of one side with 4 (4 X side). In case of rectangular shaped objects, the perimeter of a rectangle is given by adding the lengths of all four sides. Let’s see how the perimeter of a rectangle is calculated and how the formula is derived.

We learnt the definition of rectangle as a shape with four sides, and the opposite sides are equal in length and are parallel to each other. We have also covered properties of rectangle. Based on these, we can understand the perimeter of a rectangle. For any polygon (shape with any number of sides), the perimeter is calculated by adding up the length of all sides. Similarly, we calculate the perimeter of a rectangle by adding the length of all four sides. Note that the opposite sides of a rectangle are equal in length. So, if we take the length of one side as ‘a’ and that of the other side as ‘b’, then the perimeter formula can be given as 2(a) + 2(b), or 2(a+b). The longer side of a rectangle is called length and the shorter side is called breadth. So, we can also write the formula for perimeter as 2(l + b), where ‘l’ is length and ‘b’ is breadth.

Derivation of Formula of Perimeter of a Rectangle

Consider a rectangle with sides AB, BC, CD and DA. Based on definition of rectangle:

Length of sides AB and CD are equal = a

Length of sides BC and DA are also equal = b

Perimeter is equal to the sum of all sides. So, the perimeter of the rectangle will be equal to the sum of sides AB, BC, CD, and DA.

Perimeter of rectangle = AB + BC + CD + DA = a + b + a + b = 2 a + 2 b = 2(a+b)

The longer side of the rectangle is called length = a = l

The shorter side of a rectangle is breadth = b = b

So, perimeter of rectangle = 2 (length + breadth) = 2 (l + b)

Perimeter of a Square

Just like the perimeter of a rectangle, the perimeter of a square is also the total length of the border. We can find the perimeter of the square by adding up the lengths of each side. The formula for perimeter of a square is given as four multiplied by length of one side, since lengths of all sides are equal (as per definition of square).

Perimeter = 4 X side

Derivation of Formula of Perimeter of a Square

Since the perimeter of a square is given by adding the lengths of all sides, we can write the perimeter as:

Perimeter of a square = a + a + a + a (where a = length of side) = 4a

So, formula for perimeter of square (p) = 4a

There are several ways in which we can calculate the perimeter of a square. We have already seen how the perimeter can be calculated using the length of the side. Apart from this, we can also calculate the perimeter using the diagonal of the square. For this, we need to first measure the length of the diagonal, and then calculate the length of the side using the formula: Side = diagonal/√2 = √2 × diagonal/ 2. This can be derived from the properties of square. Once we know the side length, multiply by 4 and then we get the perimeter. So, formula for perimeter of a square using the length of the diagonal = Perimeter = 4(√2 × diagonal/2) = 2√2 × diagonal units.

Area of a Rectangle and Square

Area is the space occupied by a shape. Different shapes have different formulae to calculate their area. In the case of a square, we calculate the area by squaring the length of the side.

Area of a square = (Side)2

On the other hand, area of a rectangle is calculated by multiplying the length and breadth of the rectangle.

Area of a rectangle = Length X Breadth

Let’s see how the area of a rectangle and square are derived.

We learnt the definition of rectangle in an earlier section. To find the area of a rectangle, we have to multiply the length and breadth. So, the straightforward formula is Length X Breadth. But we can also calculate the area from the length of the diagonal. Similarly, area of a square can also be calculated by multiplying the length and breadth.

Area of square = Length X Breadth

But since the length and breadth are equal in a square, as per definition of square, we can simply replace the terms.

Area of square = Side X Side = Side2 = s2

FAQs on Rectangle and Squares

Question 1: Dexter has to divide his rectangular field into two parts from one corner to the other using fence. If the area of the field is 450m² and the length of the field is 36m then what will be the length of the fence needed?

Answer: Area of rectangle = length × breadth
Since the area of the field is  450m² and the length of the field is 36m the breadth would be 540/36 = 15m
The length of the fence needed is the length of the diagonal.
= √(15²) + (36²) = √225 + 1296
= √1521
= 39m

Question 2: The length of a room exceeds the breadth by 22 metres. If both the length and the breadth are increased by 1 meter, then the area of the room is increased by 11 sq. m. Find the length and the breadth of the room.

  1. 3m and 2 m
  2. 2m and 7m
  3. 7m and 9m
  4. 6m and 4m

Answer: D. Let the breadth of the room is x meter. Then,

length of room = x+ 2 (given) and
area of room =  (x+2) x sq  meter

If length and breadth increased 1 meter,
length = (x+2) + 1 = x + 3 meter and breadth = x + 1 meter
Then area of new room = ( x + 3) (x + 1)  sq m

As per given in question

( x + 3) (x + 1) –  (x+2)x = 11

= x² + 4x + 3 – x² -2x  = 11

= 2x = 8

x = 4

So breadth of room = 4

And length of room = 4 + 2 = 6

Question 3: Is a square also a rectangle?

Answer: A square shape matches the definition of rectangle. It is a type of rectangle, where all the four sides are of the same length. A square is also a rhombus, which is a four-sided shape with sides of equal length. But in rhombus, the angles can or can’t be 90 degrees. Hence, a square is a rhombus but not every rhombus is a square. This means not all rhombi have the properties of square.

Question 4: Who found the square shape?

Answer: The circle, triangle, and square are some of the basic shapes on earth. They support structures both natural and artificial. However, in the 1960s, Italian artist Bruno Munari explored the visual history of these shapes in three books which were recently compiled by Princeton Architectural Press into Bruno Munari: Square Circle Triangle.

Question 5: Are the diagonal and sides of the square equal?

Answer: No. The definition of square does not state this. Diagonal of a square is equal to the length of one of the sides of the square times 2√ or s2√. E.g. if the side is of length 8 units then the diagonal will be 8X2√.

Question 6: Is square a polygon?

Answer: In a regular four sided polygon, all the sides are of the same length and its angles are equal. Hence, a regular four-sided polygon is a square.

Question 7: What is the difference between a square and a rectangle?

Answer: A major difference between square and rectangle is that a square has four equal sides while in a rectangle, only two opposite sides are equal. Another difference based on the definition of square is that the diagonals of a square bisect each other at right angles. On the other hand, the diagonals of a rectangle also bisect each other but not at right angles.

Question 8: Is square called a special rectangle?

Answer: A square is also called a special rectangle because it possesses some properties of rectangles and fits the definition of rectangle, like interior angles being 90-degrees, opposite sides being equal and parallel, and the diagonals bisect each other. But the square has two additional features which make it a special rectangle: all four sides are equal and the diagonals bisect each other at 90-degrees.

We have now seen the definition of rectangle, definition of square, square shaped objects, rectangular shaped objects, properties of rectangle, properties of square, and the perimeters of rectangle and square, in addition to the calculation of their areas.

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