Square formula properties area and perimeter with examples
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 Understanding the Square in Geometry
1. What is a square in mathematics?
A square is a four-sided polygon with all sides equal and all interior angles equal to 90°. It is a special type of quadrilateral that is both a rectangle and a rhombus. Key properties include:
- Four equal sides
- Four right angles
- Diagonals that are equal and bisect each other at right angles
- Opposite sides are parallel
2. What is the formula for the area of a square?
The area of a square is calculated using the formula Area = side × side = s². If the side length is s, multiply the side by itself.
- Formula: A = s²
- Example: If s = 5 cm, then A = 5² = 25 cm²
3. What is the perimeter of a square?
The perimeter of a square is four times the length of one side, given by P = 4s. Since all sides are equal, add the same side four times.
- Formula: P = 4 × s
- Example: If s = 6 m, then P = 4 × 6 = 24 m
4. How do you find the diagonal of a square?
The diagonal of a square is found using the formula d = s√2, where s is the side length. This comes from applying the Pythagorean theorem to the right triangle formed by the sides.
- Formula: d = s√2
- Example: If s = 4 cm, then d = 4√2 cm
5. Why is a square both a rectangle and a rhombus?
A square is both a rectangle and a rhombus because it satisfies the properties of both shapes. A rectangle has four right angles, and a rhombus has four equal sides. A square has:
- Four equal sides (rhombus property)
- Four right angles (rectangle property)
6. What are the properties of a square?
The properties of a square include equal sides, equal angles, and special diagonal behavior. Important properties are:
- All sides are equal
- Each interior angle is 90°
- Opposite sides are parallel
- Diagonals are equal in length
- Diagonals bisect each other at right angles
7. How many lines of symmetry does a square have?
A square has 4 lines of symmetry. These lines divide the square into two identical halves.
- Two lines through the midpoints of opposite sides (vertical and horizontal)
- Two lines along the diagonals
8. What is the difference between a square and a rectangle?
The main difference between a square and a rectangle is that a square has all sides equal, while a rectangle only requires opposite sides to be equal. Comparison:
- Square: Four equal sides and four right angles
- Rectangle: Opposite sides equal and four right angles
9. How do you find the side length of a square if the area is given?
To find the side length of a square from the area, take the square root of the area using s = √A. This reverses the area formula A = s².
- Formula: s = √A
- Example: If A = 49 cm², then s = √49 = 7 cm
10. What are some real-life examples of a square?
Common real-life examples of a square include objects with four equal sides and right angles. Examples are:
- Chessboard tiles
- Floor tiles
- Square windows
- Carrom board
