What are the Properties and Formulas of a Square?
The concept of square in maths plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Every student studies squares in geometry, number theory, and even mental maths tricks for exams like NTSE, Olympiad, and CBSE boards.
What Is Square in Maths?
A square in maths is defined as a quadrilateral (a four-sided polygon) where all the sides are equal, and all the angles are right angles (90°). In other words, a square is a regular quadrilateral. You’ll find this concept applied in geometry, mensuration, and coordinate geometry.
| Property | Value |
|---|---|
| Number of sides | 4 (all equal) |
| Number of angles | 4 (all 90°) |
| Diagonals | 2 (equal length, bisect at 90°) |
| Lines of symmetry | 4 |
| Rotational symmetry | Order 4 |
| Area | side × side |
| Perimeter | 4 × side |
Key Formula for Square in Maths
Here’s the standard formula:
Area = \( s \times s \) = \( s^2 \)
Perimeter = \( 4 \times s \)
Diagonal = \( s \times \sqrt{2} \)
Properties of a Square
- All four sides are equal and parallel opposite sides.
- All four angles measure 90°.
- Diagonals are equal, bisect each other at 90°.
- Four lines of symmetry and rotational symmetry of order 4.
- Diagonals also bisect the angles of the square.
Difference Between Square and Rectangle
| Feature | Square in Maths | Rectangle |
|---|---|---|
| Sides | All sides equal | Opposite sides equal |
| Angles | All 90° | All 90° |
| Diagonals | Equal, bisect at 90° | Equal, bisect (not always at 90°) |
| Symmetry | 4 lines | 2 lines |
Read more: Difference Between Square and Rectangle
How to Construct a Square
- Draw one side AB of a desired length (let’s say 5 cm).
- At point A, use your compass and protractor to construct a 90° angle.
- Mark point D so that AD = AB.
- At point B, construct another 90° angle and mark point C so that BC = AB.
- Join points C and D. The figure ABCD is your square in maths!
To see this step visually, visit Construction of Square.
Square in Real Life
- Tiles on the floor
- Chessboard squares
- Window panes
- Post-it notes
These everyday objects make it easy to visualize and apply the concept of square in maths to the outside world.
Step-by-Step Illustration
- Find the area and perimeter of a square whose side is 6 cm.
-
Area = \( 6 \times 6 = 36 \) cm²
Perimeter = \( 4 \times 6 = 24 \) cm
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with square in maths. Many students use this trick during timed exams to save crucial seconds.
Example Trick: To square numbers ending in 5 (example: 35²):
- Multiply the tens digit by its next number:
3 × 4 = 12 - Write 25 next to it:
1225 - So, 35² = 1225.
Tricks like this aren’t just cool — they’re practical in competitive exams like NTSE, Olympiads, and even JEE. Vedantu’s live sessions include more such shortcuts to help you build speed and accuracy.
Try These Yourself
- Draw a square of side 7 cm and label all its properties.
- Find out how many small squares there are on a standard chessboard.
- Check if a quadrilateral with equal diagonals that bisect at 90° is always a square.
- Calculate the length of the diagonal of a square with a side of 9 cm.
Frequent Errors and Misunderstandings
- Mixing up squares with rectangles (remember: all sides must be equal for a square).
- Confusing diagonal formula for rectangles and squares.
- Forgetting that every angle in a square is always 90°.
Relation to Other Concepts
The idea of square in maths is closely related to rectangle, quadrilaterals, and rhombus. Mastering these properties helps with proofs, Venn diagrams, and problem-solving in higher classes.
Classroom Tip
A quick way to remember square in maths is to visualize a chessboard or tile. All sides must be equal — if even one side is shorter or longer, it’s not a square! Vedantu’s teachers often use model cutouts and digital diagrams to reinforce this during live classes.
We explored square in maths—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving questions on this topic. For a deep dive into formulas and more MCQs, check Area of Square Using Diagonal.
