Key Math Formulas Every Student Should Know

Basic math formulas constitute the foundational relations and equations that describe fundamental arithmetic, algebraic, geometric, and trigonometric properties. These formulas are indispensable for systematic problem-solving and are used consistently throughout mathematical studies, including advanced topics such as calculus and analytical geometry.

Structure and Application of Arithmetic Operation Rules (BODMAS)

The order of arithmetic operations is governed by the conventional rule known as BODMAS, which prescribes the correct sequence for simplifying expressions: Brackets, Of (i.e., powers and roots), Division, Multiplication, Addition, and Subtraction. Each operation must be performed only after resolving operations earlier in the order.

For example, to simplify $8 + 24 \div 6 \times 2$, division and multiplication precede addition. Compute $24 \div 6 = 4$, then multiply $4 \times 2 = 8$, and finally, $8 + 8 = 16$.

Expansion and Factorization of Algebraic Expressions

Key algebraic identities are foundational for both expansion and simplification of algebraic expressions. The binomial square and cube expansions are as follows:

Square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$.

For any three real numbers $a$, $b$, and $c$, the square of their sum is given by:

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.

The cube of a binomial is expanded as: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ and $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$.

Difference and sum of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

The difference of squares and higher powers yields:

$a^2 - b^2 = (a - b)(a + b)$, $a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) = (a - b)(a + b)(a^2 + b^2)$.

Similarly, $a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$.

For symmetric expressions, $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz$, and the identity $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$ is useful in polynomial factorisation.

Application of these algebraic formulas is illustrated in the expansion of $(2x - 3)^3$ as follows:

Let $a = 2x$ and $b = 3$.
$(2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3$
$= 8x^3 - 3 \cdot 4x^2 \cdot 3 + 3 \cdot 2x \cdot 9 - 27$
$= 8x^3 - 36x^2 + 54x - 27.$

For further details on sequences and their properties, see the Arithmetic Sequence Formula reference guide.

Formulation of Geometry Perimeter, Area, and Volume Equations

The perimeter, area, and volume of standard geometric figures are given by specific formulas derived from their definitions.

Perimeter of a square: $P = 4a$, where $a$ is the length of the side.

Perimeter of a rectangle: $P = 2(l + b)$, where $l$ is the length and $b$ is the breadth.

Area of a square: $A = a^2$.

Area of a rectangle: $A = l \times b$.

Area of a triangle (with base $b$ and height $h$): $A = \frac{1}{2} b h$.

The derivation for the area of a circle must proceed explicitly:

Given a circle with radius $r$.

The circumference is $2\pi r$. Consider dissecting the circle into $n$ equal sectors and rearranging them alternately, forming an approximate parallelogram of base $\pi r$ and height $r$.

The area is then approximated as $\pi r \times r = \pi r^2$. As $n \rightarrow \infty$, the approximation becomes an equality:

Area of a circle: $A = \pi r^2$.

For details on related figures, see the Area of a Circle Formula page.

Circumference of a circle: $C = 2\pi r$.

For cuboids and three-dimensional figures:

Surface area of a cube: $S = 6a^2$.

Volume of a cube: $V = a^3$.

Volume of a cuboid: $V = l \times b \times h$.

Volume of a cylinder: $V = \pi r^2 h$, where $r$ is the radius and $h$ the height.

Curved surface area of a cylinder: $2\pi rh$.

Total surface area of a cylinder: $2\pi r(r + h)$.

Curved surface area of a cone: $\pi r l$, where $l$ is the slant height.

Volume of a cone: $V = \frac{1}{3} \pi r^2 h$.

Surface area of a sphere: $S = 4\pi r^2$.

Volume of a sphere: $V = \frac{4}{3}\pi r^3$.

For additional information on areas of quadrilaterals, visit the Area Formula for Quadrilateral resource.

Critical Relations in Percentage, Profit, and Interest Calculations

Percentage: For value $x$ as a percentage of $y$, the formula is $\left( \frac{x}{y} \right) \times 100$.

Profit percentage: $\frac{\text{Profit}}{\text{Cost Price}} \times 100$.

Simple Interest: $I = \frac{P \times R \times T}{100}$, where $P$ is principal, $R$ is rate of interest per annum, $T$ is the time in years.

The Amount after $T$ years at simple interest is $A = P + I$.

For Compound Interest compounded yearly:

$A = P \left(1 + \frac{R}{100}\right)^n$, where $n$ is number of years.

Compound Interest $= A - P$.

Laws of Exponents and Indices in Calculations

The following exponent rules are fundamental in algebra:

Product Law: $a^m \times a^n = a^{m+n}$.

Quotient Law: $\frac{a^m}{a^n} = a^{m-n}$ for $a \neq 0$.

Zero Exponent: $a^0 = 1$, $a \neq 0$.

Negative Exponent: $a^{-n} = \frac{1}{a^n}$, $a \neq 0$.

Power of a Power: $(a^m)^n = a^{mn}$.

Power of a Product: $(ab)^n = a^n b^n$.

Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, $b \neq 0$.

Key Relations in Probability and Statistics Fundamentals

Probability of an event $A$ is $P(A) = \frac{n(A)}{n(S)}$, where $n(A)$ is the number of favourable outcomes, $n(S)$ is the size of the sample space.

Arithmetic mean (average) for $n$ values $x_1, x_2, \ldots, x_n$ is $\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}$.

For additional details on means in statistics, see the comprehensive guide on Arithmetic Mean.

Properties of Rational Numbers and Basic Equations

The additive inverse of a rational number $\frac{a}{b}$ $(b \neq 0)$ is $-\frac{a}{b}$, since $\frac{a}{b} + \left( -\frac{a}{b} \right) = 0$.

The multiplicative inverse of $\frac{a}{b}$ $(a \neq 0, b \neq 0)$ is $\frac{b}{a}$, since $\frac{a}{b} \times \frac{b}{a} = 1$.

An equation in one variable, such as $2x + 3 = 7$, asserts equality between two algebraic expressions involving an unknown.

Distributive and Other Foundational Properties

The distributive law of multiplication over addition is $a(b + c) = ab + ac$, and over subtraction is $a(b - c) = ab - ac$.

For geometric applications involving regular polygons, the area of an equilateral triangle with side $a$ is derived step-by-step:

The altitude $h$ is found using the Pythagorean theorem:
$h^2 + \left(\frac{a}{2}\right)^2 = a^2$
$h^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}$
$h = \frac{a\sqrt{3}}{2}$
Area $= \frac{1}{2} \times a \times h = \frac{1}{2} \times a \times \frac{a\sqrt{3}}{2} = \frac{\sqrt{3}}{4} a^2$.

For full derivations of formulas for equilateral triangles, visit Area of Equilateral Triangle Formula.

Summary of Critical Algebraic and Geometric Forms

Fundamental math formulas are required for every level of mathematics, including school examinations and competitive entrance exams. Greater proficiency is achieved by regular practice in applying these relations to textbook problems and real-life scenarios. Mastery of these standard equations significantly accelerates both accuracy and speed in mathematical reasoning.

For additional proofs, charts, and downloadable references on key equations, consult the Basic Math Formulas main reference resource.