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.
FAQs on Square in Maths: Meaning, Properties & Examples
1. What is a square in Maths?
In geometry, a square is a two-dimensional shape defined as a quadrilateral (a four-sided polygon) with four equal-length sides and four equal angles, each measuring 90 degrees (right angles). It is a special type of both a rectangle and a rhombus.
2. What are the main properties of a square?
The key properties that define a square are:
- All four sides are equal in length.
- All four interior angles are right angles (90°).
- The opposite sides are parallel to each other.
- The diagonals are equal in length.
- The diagonals bisect each other at a right angle (90°).
- The diagonals also bisect the corner angles, creating two 45° angles at each vertex.
3. How do you calculate the area and perimeter of a square?
You can calculate the area and perimeter of a square using its side length, 's':
- Area: The area is the space it covers, calculated by the formula Area = s² (side multiplied by side).
- Perimeter: The perimeter is the total length of its boundary, calculated by the formula Perimeter = 4s (4 times the side length).
4. What are some real-life examples of a square shape?
Square shapes are common in everyday objects. Some examples include:
- A single square on a chessboard or checkerboard.
- The face of a standard six-sided dice or a Rubik's cube.
- Certain types of floor tiles or window panes.
- A square-shaped coaster or a piece of origami paper.
5. What is the difference between a square, a rectangle, and a rhombus?
A square is a more specific version of both a rectangle and a rhombus.
- Square vs. Rectangle: A rectangle needs four right angles, but only its opposite sides must be equal. A square also has four right angles, but with the stricter condition that all four sides must be equal.
- Square vs. Rhombus: A rhombus needs four equal sides, but its angles don't have to be 90°. A square also has four equal sides, but with the stricter condition that all four angles must be right angles.
6. Why is the diagonal of a square useful to know?
The diagonal of a square (a line connecting opposite corners) is very useful because it helps you find other measurements, especially when the side length isn't given. If you know the diagonal 'd', you can directly calculate the Area = d²/2. This formula is derived from the Pythagorean theorem and provides a shortcut for solving problems.
7. How is 'squaring a number' in algebra related to a geometric square?
The mathematical term "squaring a number" (like 5²) originates from the geometry of a square. To find the area of a square, you multiply its side length by itself (side × side). This operation, s × s, is written as s². Therefore, multiplying any number by itself is called "squaring" because it represents the process of finding the area of a geometric square whose side length is that number.
8. Why is a square considered a regular polygon?
A polygon is defined as "regular" if all of its sides are equal in length and all of its interior angles are equal in measure. A square perfectly meets these two conditions: it has four equal sides and four equal angles (each 90°). This is why a square is classified as a regular polygon, specifically a regular quadrilateral.
