1. Definition of a Square
A square is a quadrilateral in which all four sides are equal and all four angles are right angles (each measuring \(90^\circ\)).
In simple words, a square is both a rectangle (because it has right angles) and a rhombus (because all sides are equal). This makes it the most perfect form of a parallelogram.
2. Properties of a Square
A square has many geometric properties that make calculations easy and predictable. It is often used in geometry, mensuration, and coordinate geometry.
2.1. All Sides are Equal
If the side of a square is \(a\), then:
\(AB = BC = CD = DA = a\)
2.2. All Angles are Right Angles
Each interior angle is:
\(90^\circ\)
This makes the square a right-angled quadrilateral.
2.3. Opposite Sides are Parallel
Even though all sides are equal, opposite sides still remain parallel:
\(AB \parallel CD\)
\(BC \parallel AD\)
2.4. Diagonals are Equal
The diagonals of a square have equal length:
\(AC = BD\)
2.5. Diagonals are Perpendicular
The diagonals meet at right angles:
\(AC \perp BD\)
2.6. Diagonals Bisect Each Other
The diagonals cut each other into two equal halves at their intersection point.
3. Perimeter and Area of a Square
Because all sides are the same, perimeter and area formulas become very simple.
3.1. Perimeter
If each side is \(a\), then:
\(\text{Perimeter} = 4a\)
3.2. Area
The area of a square is:
\(\text{Area} = a^2\)
3.3. Quick Example
Side of a square = \(6\text{ cm}\)
\(\text{Area} = 6^2 = 36\text{ cm}^2\)
4. Diagonals of a Square
The diagonals of a square are very useful in coordinate geometry, mensuration, and proving geometric results.
4.1. Diagonal Length Formula
Using Pythagoras theorem on the right triangle formed by two sides and the diagonal:
\(d = a\sqrt{2}\)
4.2. Example
If \(a = 10\text{ cm}\), then diagonal:
\(d = 10\sqrt{2}\text{ cm}\)
5. Square as a Special Quadrilateral
A square belongs to multiple quadrilateral families at the same time because it satisfies all their properties.
5.1. Square as a Rectangle
All angles are \(90^\circ\). The difference is that in a square, all sides are equal.
5.2. Square as a Rhombus
All sides are equal. The difference is that in a square, all angles are right angles.
5.3. Square as a Parallelogram
Opposite sides are equal and parallel, and diagonals bisect each other—just like in a parallelogram.