Class VIII Mathematics - Formulae & Properties

A comprehensive guide to mathematical identities, properties, and mensuration formulas.

Chapter 1: Rational Numbers

• Associative Property of Addition: For any three rational numbers $a$, $b$, and $c$:
  \(a + (b + c) = (a + b) + c\)
• Associative Property of Multiplication: For any three rational numbers $a$, $b$, and $c$:
  \(a \times (b \times c) = (a \times b) \times c\)
• Multiplicative Identity: $1$ is the identity for multiplication.
  \(1 \times a = a \times 1 = a\)
• Distributive Property: \(a(b + c) = ab + ac\) and \(a(b - c) = ab - ac\)

Chapter 3: Understanding Quadrilaterals

• Sum of Interior Angles of a Polygon: For a polygon with $n$ sides:
  \((n - 2) \times 180^\circ\)
• Sum of Measures of Exterior Angles: For any convex polygon, the sum is always \(360^\circ\).
• Angle Sum of a Quadrilateral: The sum of the four interior angles is \(360^\circ\).

Chapter 9: Algebraic Expressions and Identities

• Identity I: \((a + b)^2 = a^2 + 2ab + b^2\)
• Identity II: \((a - b)^2 = a^2 - 2ab + b^2\)
• Identity III: \((a + b)(a - b) = a^2 - b^2\)
• Identity IV: \((x + a)(x + b) = x^2 + (a + b)x + ab\)

Chapter 11: Mensuration

• Area of a Trapezium: \(\frac{1}{2} \times (\text{sum of parallel sides}) \times h\)
• Area of a Rhombus: \(\frac{1}{2} \times d_1 \times d_2\), where $d_1, d_2$ are diagonals.
• Surface Area of a Cube: \(6l^2\), where $l$ is the side length.
• Total Surface Area of a Cuboid:
  \(2(lb + bh + hl)\)
  ($l$=length, $b$=breadth, $h$=height)
• Volume of a Cylinder: \(\pi r^2 h\)

Chapter 12: Exponents and Powers

• Product Law: \(a^m \times a^n = a^{m+n}\)
• Quotient Law: \(a^m \div a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{mn}\)
• Same Exponent: \(a^m \times b^m = (ab)^m\)
• Zero Power: \(a^0 = 1\) (where \(a \neq 0\))
• Negative Exponent: \(a^{-m} = \frac{1}{a^m}\)

Chapter 13: Direct and Inverse Proportions

• Direct Proportion: $\frac{x}{y} = k$ (Constant). As $x$ increases, $y$ increases proportionally.
• Inverse Proportion: $x \times y = k$ (Constant). As $x$ increases, $y$ decreases proportionally.