Square: Definition और Properties

Definition and Properties

• A square is a quadrilateral with four equal sides and four right angles (90 degrees).
• All sides are congruent (equal in length).
• All angles are congruent (equal in measure).
• Opposite sides are parallel.
• Diagonals are congruent.
• Diagonals bisect each other at right angles.
• Diagonals bisect the angles (each diagonal divides the four angles into halves).

Area and Perimeter

• Area: The area of a square is calculated by squaring the length of one side (Area = side²).
• Perimeter: The perimeter of a square is calculated by multiplying the length of one side by 4 (Perimeter = 4 * side).

Relationships to Other Shapes

• A square is a special type of rectangle.
• A square is a special type of parallelogram.
• A square is a special type of rhombus.
• A square is a regular polygon (all sides and angles are equal).

Diagonals

• The length of a diagonal of a square with side length 's' is s√2.
• The two diagonals of a square intersect at a right angle (90 degrees), creating four congruent right isosceles triangles.

Special Cases and Applications

• A square is a very fundamental geometric figure.
• Found in many constructions.
• Used as a basic model for many design and technical problems (architecture, engineering).
• A fundamental concept for understanding other geometric shapes.

Connections to other mathematical concepts

• The concept of a square is essential in algebra, trigonometry, and calculus.
• Squares play a fundamental role in understanding geometry in general.
• A square is a fundamental element in understanding coordinate systems and functions in higher mathematics.
• Squares give rise to concepts of symmetry, rotations, and reflections in geometry.

Visual Representation

• A square is a closed two-dimensional shape with four equal straight sides connected to create a polygon/quadrilateral with a regular/symmetrical form.

Other Important Characteristics

• The center of a square is the intersection of its diagonals.
• The center of rotation is at the center of intersection of diagonals.