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Square in Maths – Area, Perimeter, Examples & Applications

Last Updated : 03 May, 2024
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Square is a type of quadrilateral with four sides. What distinguishes a square from other quadrilaterals is that all four sides of a square are of equal length, and all four interior angles are right angles (90 degrees).

Let’s learn about Square, including its properties, area, perimeter, examples, practice problems, and applications of Square in real life.

What is a Square?

A square is a geometric shape with four equal sides and four right angles. Each side of a square is parallel to the opposite side, and all angles are 90 degrees. In a square, the diagonals are equal in length and bisect each other at right angles. Squares are a type of quadrilateral, which is a polygon with four sides. They have various applications in mathematics, architecture, engineering, and everyday life.

Square Definition

Square is a quadrilateral whose all four sides are equal and measure of each angle is 90°.

A square is made up of two pairs of parallel lines. All four sides of a square are equal. Its opposite sides are parallel to each other and the interior angles are equal to 90°.

Square Diagram

Square Shape

A square is a type of four-sided shape known as a polygon. What makes a square unique is that all of its sides have the same length, and all of its angles measure exactly 90 degrees. When you cut a square perfectly in half using a plane that passes through its center, both resulting halves will be mirror images of each other.

Square Examples

A square is a very common shape and can be seen in a variety of objects which we use in our daily lives. Various Square-shaped objects are chess boards, carrom boards, Ludo, etc.

Examples of Square

Properties of Square

Some general properties of the square are :

  • Square is a quadrilateral with 4 sides and 4 vertices.
  • All four sides of the square are equal.
  • Opposite pairs of sides of a square are parallel to each other.
  • Each interior angle of a square is 90°.
  • Diagonals of a square are perpendicular bisectors of each other.
  • Diagonals of a square are of equal length.
  • Sum of all interior angles of a square is 360°.
  • Diagonals of a square divide it into two congruent triangles.

Square Formulas

We know that a square is a four-sided figure with equal sides. There are three basic square formulas that are commonly used in geometry for squares are,

  • Area of Square
  • Perimeter of Square
  • Diagonal of Square

Let us learn these square formulas in detail.

Area of Square

Area of a square is defined as the total space occupied inside its boundaries. The formula for calculating the area of a square with sides ‘a’ is given by the formula,

Area of Square = a2 

Area of Square (When Diagonal is Given)

If the diagonal of a square is given then the area of the square is given by 

Area = d2/2

Perimeter of Square

Perimeter of a square is defined as the length of all its boundaries. Suppose the length of the sides of a square is ‘a’ then its perimeter is given by

Perimeter of Square = Sum of All Sides of Square = a + a + a + a 

Perimeter of Square = 4a units

As a result, 4a is perimeter of a square with each side’s length equal to ‘a’ units.

Diagonal of Square

Diagonals of square is equal to a√2, where a is the side of the square. The length of both diagonals of a square is equal to each other. The relation between diagonals and sides of a square is given by Pythagoras Theorem.

Diagonal of Square

Length of Diagonal of Square

Length of diagonal of a square is calculated using Pythagoreas Theorem as,

Hypotenuse2 = Base2 + Perpendicular2

Hence, 

Diagonal2 = Side2 + Side2

d2 = s2 + s2

d2 = 2s2

d = s√2

Construction of Square

Square in geometry can be constructed using the steps added below,

For example, Construct asquare with sides 4 cm.

Step 1: Draw a line segement AB of 4 cm

Step 2: From point A and B draw two right angles.

Step 3: Take an arc of 4 cm and mark on right angles from A and B and mark these points C and D respectively.

Step 4: Join CD, ABCD is the required square.

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Applications of Squares in Real Life

  • Floor tiles often come in square shapes, providing a uniform and tidy appearance.
  • Many rooms in buildings have square or rectangular layouts, making them easier to design and construct.
  • Squares are fundamental to geometry and algebra, used in various calculations and equations.
  • Digital images are composed of square pixels arranged on a grid, forming pixel art.
  • Screens of electronic devices like computer monitors and smartphones are often square or rectangular.
  • Integrated circuits, the building blocks of electronic devices, are typically manufactured on square silicon wafers.
  • Chess and checkers boards feature square grids, offering a strategic playing field.
  • Many packages and storage containers are square or rectangular for efficient stacking and storage.
  • Books generally have square or rectangular pages, making them easy to store on shelves and in boxes.
  • Agricultural fields are often laid out in square or rectangular shapes for efficient land use and irrigation.
  • Many sports fields and courts, including those for soccer, football, basketball, and tennis, have square or rectangular boundaries.
  • Board games like Scrabble, Sudoku, and crossword puzzles often use square grids or boards.

Examples on Squares

Some Examples on Square Formulas are,

Example 1: A square has one of its sides measuring 24 cm. Calculate its area and perimeter.

Solution: 

Given,

  • Side of Square = 24 cm

Area of Square = a2

Area of Square = 24 × 24 = 576 sq cm 

Perimeter of Square =  Sum of all sides of square = a + a + a + a = 4a 

P = 4 × 24 

P = 96 cm

Hence, area of square is 576 sq. cm and perimeter of square is 96 cm.

Example 2: Find the area of a square park whose perimeter is 420 ft.

Solution:   

Given,

  • Perimeter of Square Park = 420 ft

Perimeter of a Square = 4 × side

4 × side = 420

Side = 420/4

Side = 105 ft

Formulae for Area of a Square = side2

Hence, Area of Square Park = (105)2 

A = 105 × 105 = 11025 ft2

Thus, area of a square park whose perimeter is 420 ft is 11025 ft2

Example 3: A rectangular floor is 40 m long and 30 m wide. Square tiles, each of 4 m side lengths, are to be used to cover the floor. What will be the total number of tiles which will be required to cover the floor?

Solution: 

Given,

  • Length of Floor = 40 m
  • Breadth of Floor = 30 m

Area of Rectangular Floor = length × breadth = 40 m × 30 m = 1200 sq. m

Side length of one tile = 4 m

Area of one tile = side × side = 4 m × 4 m = 16 sq. m

Now number of tiles needed = Total Area of Floor / Area of One Tile = 1200 /16 = 75 tiles 

Total 75 tiles are needed to cover the area of rectangular floor of 1200 sq m.

Example 4: Find the area of a square park whose diagonal is 15 m.

Solution: 

Given,

  • Diagonal of Square Park = 15 m

Area of a Square Formula when Diagonal is Given = (d)2 / 2 

Hence, Area of Square Park = (15)2/ 2

A = (15 × 15) / 2 = 225 / 2 = 112.5 sq. m

Thus, area of a square park whose diagonal is 15 m is 112.5 sq. m.

Practice Questions on Square

Various practice questions on square are,

Q1: Find the perimeter of a square of its side 24 in.

Q2: If the side of a square chess board is 120cm in, then find its perimeter.

Q3: What is area and perimeter of a square if its side is 15 cm?

Summary – What is Square

Squares are four-sided shapes where all sides are equal, and all angles are 90 degrees. They’re found in various objects like chessboards, rooms, and electronic screens. Key properties include equal sides, parallel opposite sides, and perpendicular bisecting diagonals. Basic formulas for squares include area (side squared), perimeter (4 times the side length), and diagonal length (side length times square root of 2). Squares are everywhere in real life, from floor tiles to electronic screens to board games. They’re preferred for their simplicity, symmetry, and ease of measurement.

FAQs on What is Square

What is a Square in Maths?

A square is a polygon, with all sides equal and opposite sides parallel to each other. Angles in a square are equal and the measure of each angle is 90 degrees.

What is Area of Square Formula?

Area of a Square is, Area = (side)2

What is Perimeter of Square Formula?

Perimeter of a Square is, Perimeter = 4 x side.

Can Rhombus be considered a Square?

No, a rhombus is not a square, but every square can be considered a rhombus.

Is Square a Polygon?

Square is a closed figure formed by four straight lines So it is considered a quadrilateral. Every quadrilateral is a polygon. So a square is a polygon.

Is Square Different From Rectangle?

Yes, a square is different from Rectangle. In a square, all its sides are equal whereas in a rectangle only its opposite sides are equal.

How is a Square Different from a Rhombus?

Square and Rhombus both are Equilateral Quadrilaterals, i.e. both have equal all four sides. The difference between them is that all angles in a square are equal and right angles, but on the other hand all the angles of a rhombus need not be equal.

Hence, Rhombus with right angles is called a Square. Thus, “Every Square is a Rhombus but all Rhombus are not Squares.”



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