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:
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- 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:
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- 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:
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- 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).