Square

What is a Square?

A square is a two-dimensional shape that belongs to the quadrilateral family, which means it has four sides. A square is special because all four sides are of equal length, and all four angles are right angles (90 degrees).

 

In simple words:

A square has 4 equal sides.  

A square has 4 equal angles, each measuring 90°.  

A square is a regular quadrilateral, meaning all sides and all angles are the same.  

A square is also a special type of rectangle (which has 4 right angles) and a rhombus (which has 4 equal sides). You can think of a square as a combination of both a rectangle and a rhombus.

 

Table of Content

• Properties of a Square
• Area and Perimeter of a Square
• Diagonal of a Square
• Square in Coordinate Geometry
• Real-Life Examples of Squares
• Activities for Practice
• Summary Table
• Conclusion
• FAQs on Square

Properties of a Square

The square is a highly symmetric shape. Here are its most important properties:

All four sides are equal in length.  

Example: If one side is 6 cm, the remaining three sides are also 6 cm.

 

All four interior angles are 90°.  

This means each corner forms a perfect right angle.

 

Opposite sides are parallel.  

Line AB is parallel to line CD, and line AD is parallel to line BC.

 

Diagonals are equal in length.  

The two diagonals of a square are the same length.

 

Diagonals bisect each other at 90°.  

This means the diagonals cut each other in half and form a right angle where they cross.

 

4 Lines of symmetry.  

You can fold a square along its diagonals and midlines, and both halves will match perfectly.

 

Rotational symmetry of 90°, 180°, 270°, and 360°.  

If you rotate a square by 90 degrees, it still looks the same. This is true for 180°, 270°, and 360° too.

These properties make the square a very useful shape in mathematics and design.

 

Area and Perimeter of a Square

We use two main formulas to calculate a square’s measurements: area and perimeter.

Formula for Area of a Square:  

Area = side × side  

Or simply,  

Area = side²  

Example:  

If side = 4 cm,  

Area = 4 × 4 = 16 cm²  

This tells us how much space the square covers on a flat surface like a tile or table.

 

Formula for Perimeter of a Square:  

Perimeter = 4 × side  

Example:  

If side = 4 cm,  

Perimeter = 4 × 4 = 16 cm  

The perimeter is the distance around the outside of the square, which is useful for measuring borders or fences.

 

Diagonal of a Square

The diagonal of a square is a straight line that connects opposite corners (vertices).

To calculate the diagonal, we use the Pythagorean Theorem, since the square can be divided into two right-angled triangles.

Formula for Diagonal of a Square:  

Diagonal = side × √2  

Example:  

If side = 5 cm,  

Diagonal = 5 × √2 ≈ 7.07 cm  

The diagonal helps in splitting the square into two equal right-angled triangles, which is useful in both geometry and construction.

 

Square in Coordinate Geometry

In coordinate geometry, we can draw and prove squares using points (x, y) on a graph.

To draw a square:  

Plot 4 points so that the distance between each side is equal.  

Use the distance formula to verify side lengths.  

Use the slope formula to prove that all angles are 90°.

Distance Formula:  

Distance = √((x2 − x1)² + (y2 − y1)²)

 

Midpoint Formula (to check if diagonals bisect each other):  

Midpoint = ((x1 + x2)/2 , (y1 + y2)/2)

 

Slope Formula (to check perpendicular diagonals):  

Slope = (y2 − y1)/(x2 − x1)  

By using these formulas, you can prove whether a given shape on the coordinate plane is a square.

 

Real-Life Examples of Squares

Squares are all around us. Learning to recognize them in daily life builds your visual and spatial skills.

Common Examples:  

• Tiles on the floor - most are squares for uniformity.  
• Chessboard - made of 64 small squares.  
• Windows and doors - often square or rectangular.  
• Paper napkins and handkerchiefs - usually square in shape.  
• Sticky notes - standard size is square.  
• Books and boards - some are square in layout.  

In architecture, construction, fashion, and graphic design, square shapes are widely used for symmetry and efficiency.

 

Activities for Practice

Try these activities to build your understanding:

Draw squares with sides of 3 cm, 5 cm, and 7 cm.  

Calculate area and perimeter for each.  

Use graph paper to plot a square using coordinates.  

Fold a square paper to find all 4 lines of symmetry.  

Measure diagonals and confirm using the formula: Diagonal = side × √2.  

Color a chessboard to understand how many small squares it has.  

 

Summary Table

 

Conclusion

The square is one of the most important and easy-to-recognize shapes in geometry. Its perfect symmetry, equal sides, and angles make it ideal for both learning and real-life use. Whether you're measuring space, identifying patterns, or designing objects, knowing how to work with squares helps build a solid foundation in math. Mastering squares is also your stepping stone to learning about rectangles, rhombuses, parallelograms, and more geometric concepts.

 

Related Links

Parallelograms - Explore key properties, formulas, solved examples, and real-life applications.
Quadrilaterals - From trapeziums to rectangles - discover types, rules, and practice questions.

 

Frequently Asked Questions on Square

1. What is a square in geometry?

Ans: In geometry, a square is a 2D closed shape (quadrilateral) that has four equal sides and four right angles (90° each). It is a type of regular polygon, meaning all sides and angles are equal. A square is also a special kind of rectangle (because of right angles) and rhombus (because of equal sides).

 

2. What is square and its formula?

Ans: A square is a shape with:

• 4 equal sides

• 4 equal angles (each 90°)

• Equal diagonals that bisect each other at right angles

Main formulas for a square:

• Area = side × side = side²

• Perimeter = 4 × side

• Diagonal = side × √2

These formulas help calculate surface area, border length, and inner diagonal length.

 

3. What are the 7 properties of a square?

Ans: Here are 7 key properties of a square:

1. All four sides are equal in length

2. All four angles are 90 degrees (right angles)

3. Opposite sides are parallel

4. Diagonals are equal in length

5. Diagonals bisect each other at 90°

6. It has 4 lines of symmetry

7. It has rotational symmetry at 90°, 180°, 270°, and 360°
   
   

4. How can we define a square?

Ans: A square is defined as a regular quadrilateral with four equal sides and four right angles. It is a flat shape with high symmetry, and both its diagonals are equal in length and intersect at right angles. A square can also be described as a shape that is both a rectangle and a rhombus at the same time.

 

Dive into exciting math concepts, learn and explore with Orchids International School!