Understanding the Square: Properties, Formulas, and Real-Life Examples

Square: Detailed Explanation

A square is a two-dimensional geometric figure that falls under the category of quadrilaterals, specifically parallelograms. Below are the key properties and details of a square:

Properties of a Square:

All sides are equal: In a square, all four sides have the same length.

All angles are right angles: Each of the four angles in a square is 90 degrees.

Opposite sides are parallel: Like other parallelograms, a square has opposite sides that are parallel.

Diagonals:

    • The diagonals of a square are equal in length.
    • The diagonals bisect each other at right angles (90 degrees).
    • The diagonals also bisect the internal angles of the square (cutting each 90-degree angle into two 45-degree angles).

Area and Perimeter:

    • Area = Side² (If the side of a square is ‘a,’ the area is a²).
    • Perimeter = 4 × Side (If the side of a square is ‘a,’ the perimeter is 4a).

Symmetry:

    • A square has four lines of symmetry.
    • It is also rotationally symmetric by 90°, 180°, 270°, and 360°.

Comparison of Square with Other Figures:

Figure Similarities with Square Differences from Square
Rectangle – Opposite sides are equal and parallel.
– All angles are 90°.
– Only opposite sides are equal, not all sides like in a square.
– Diagonals do not bisect each other at 90°.
Rhombus – All sides are equal.
– Opposite sides are parallel.
– The angles are not necessarily 90°.
– Diagonals bisect each other at 90°, but they are not equal.
Parallelogram – Opposite sides are parallel and equal. – The angles are not 90°.
– Diagonals do not bisect each other at 90°, and they are not equal.
Trapezium – Both are quadrilaterals (four-sided shapes). – Only one pair of opposite sides is parallel.
– Sides and angles are not equal in a trapezium.

Square vs Rectangle:

Commonality: Both squares and rectangles have 90° angles at each corner, and opposite sides are parallel.

Difference: In a rectangle, only the opposite sides are equal, whereas in a square, all four sides are equal.

Square vs Rhombus:

Commonality: Both squares and rhombuses have all sides of equal length.

Difference: A square has all right angles, while a rhombus has slanted angles. The diagonals of a rhombus bisect each other at 90° but are not of equal length, unlike in a square.

Real-Life Examples of a Square:

Chessboard/Checkerboard: The individual squares on the board are perfect examples of squares.

Tiles: Many floor tiles are square-shaped.

Paper: Origami or certain sheets of paper used in crafting are square-shaped.

Important Formulas:

Area of a square = Side × Side = Side².

Perimeter of a square = 4 × Side.

Diagonal of a square = a2a\sqrt{2} (where ‘a’ is the length of the side).

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