Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Geometry → Understanding Quadrilaterals → Types of Quadrilaterals ↓


Square


In geometry, the square is one of the simplest and most symmetrical types of quadrilaterals. Understanding the properties and characteristics of the square can help in understanding more complex concepts in geometry. The square is a special type of quadrilateral that is equilateral and right-angled, meaning that all sides are the same length and all angles are the same size.

Definition of class

A square is a four-sided polygon, known as a quadrilateral, where:

  • All the four sides are of equal length.
  • All four angles are right angles (90 degrees).
  • The opposite sides are parallel.

Mathematically, a square can be defined using certain formulas and properties, with each side represented by s.

Properties of the square

Some of the important properties of class are as follows:

  • Equal sides: All four sides of a square are of equal length.
  • Right angle: Each angle in a square is 90 degrees.
  • Diagonal relation: A square has two diagonals and their lengths are equal. Each diagonal bisects the angle through which it is drawn.
  • Symmetry: A square has four lines of symmetry and rotational symmetry of order 4.

Mathematical Formulas

The perimeter and area of a square are calculated as follows:

Circumference

    If s is the length of a side of a square, then:
    Perimeter, P = 4s.

Area

    The area A of a square is given by:
    A = s².

Diagonals

    The length of each diagonal in a square, d, can be calculated as follows:
    d = s√2.

Visual Example

Consider a square with one side inscribed s:

S S S S

Each side of the square is marked s, and the diagonals intersect each other at the centre.

Understanding through examples

Example 1

If each side of the square is 5 cm, find the perimeter.

    Solution:
    Use the formula for the perimeter P = 4s.
    Here, s = 5 cm.
    Hence, P = 4 × 5 = 20 cm.

Example 2

Given a square with an area of 100 sq. cm, calculate the length of one of its sides.

    Solution:
    Use the formula for area A = s².
    Given A = 100 cm².
    So, s² = 100.
    Therefore, s = √100 = 10 cm.

Applications of classes

Squares are used in a variety of real-life contexts and mathematical applications:

  • Tiling and Flooring: Square tiles are used in building construction.
  • Board games: Games like chess and Sudoku have boards made up of square grids.
  • Computer graphics: Pixels in digital images are usually square.

Interesting facts about squares

Understanding squares also reveals some interesting mathematical facts:

  • A special example is a square, rhombus and rectangle.
  • The diagonals of a square not only bisect each other but also intersect at right angles (90 degrees).
  • Square can be found as the largest quadrilateral inside the circle.

Conclusion

The square is a fundamental shape in geometry that has unique and easily recognizable properties. Knowing how to recognize and calculate its dimensions is important for solving a variety of mathematical problems. Because of their symmetry and simplicity, squares are ubiquitous in both theoretical mathematics and practical applications.


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Square

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