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Mathematics is built on formulas that simplify problem-solving and help in quick calculations. Each branch—algebra, geometry, mensuration, trigonometry, probability, etc.—has its own set of formulas that are used frequently in academics, competitive exams, and practical life.

Algebra Formulas

Various algebraic formulas that are widely used:

Basic Algebraic Formulas

1. $a^2 - b^2 = (a - b)(a + b)$
2. $(a + b)^2 = a^2 + 2ab + b^2$
3. $a^2 + b^2 = (a + b)^2 - 2ab$
4. $(a - b)^2 = a^2 - 2ab + b^2$
5. $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
6. $(a - b - c)^2 = a^2 + b^2 + c^2 - 2ab + 2bc - 2ca$
7. $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
8. $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
9. $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
10. $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
11. $(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$
12. $(a - b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4$
13. $a^4 - b^4 = (a - b)(a + b)(a^2 + b^2)$
14. $(a^m)(a^n) = a^{m + n}$
15. $(ab)^m = a^m b^m$
16. $(a^m)^n = a^{mn}$

Mensuration Formulas

Mensuration is the study of areas and volumes of 2D and 3D shapes using mathematical formulas.

2D Shapes

Rectangle

• Perimeter of Rectangle = $2(l + b)$
• Area of Rectangle = $l \times b$

Square

• Area of Square = $a^2$
• Perimeter of Square = $4a$

Triangle

• Area of Triangle = $\frac{1}{2} \times b \times h$

Trapezoid

• Area of Trapezoid = $\frac{1}{2} \times (b_1 + b_2) \times h$

Circle

• Area of Circle = $\pi r^2$
• Circumference of Circle = $2\pi r$

3D Shapes

Cube

• Surface Area of Cube = $6a^2$
• Volume of Cube = $a^3$

Cylinder

• Curved Surface Area of Cylinder = $2\pi rh$
• Total Surface Area of Cylinder = $2\pi r(r + h)$
• Volume of Cylinder = $\pi r^2 h$

Cone

• Curved Surface Area of Cone = $\pi rl$
• Total Surface Area of Cone = $\pi r(r + l) = \pi r[r + \sqrt{h^2 + r^2}]$
• Volume of Cone = $\frac{1}{3} \times \pi r^2 h$

Sphere

• Surface Area of a Sphere = $4\pi r^2$
• Volume of a Sphere = $\frac{4}{3} \times \pi r^3$

Probability Formula

$P(A) = \frac{n(A)}{n(S)}$

Where:

• P(A) is the Probability of an Event
• n(A) is the Number of Favorable Outcomes
• n(S) is the Total Number of Events

Fraction Formulas

A fraction is a number expressed with integers in which the numerator is divided by the denominator. A fraction is basically the quotient of a division.

Addition of a whole number and a fraction

$\left(a + \frac{b}{c}\right) = \frac{(a \times c) + b}{c}$

Addition of fractions with the same denominator

$\frac{a}{b} + \frac{d}{b} = \frac{a + d}{b}$

Addition of fractions with different denominators

$\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c}{b \times d}$

Multiplication of fractions

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Division of fractions

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Percentage Formula

A percentage is a numerical value or ratio expressed as a fraction of 100. It is generally symbolized by the sign %.

$\text{Percentage} = \frac{\text{Given Value}}{\text{Total Value}} \times 100$

Distance Formula

If the coordinates of points A are $(x_1, y_1)$ and B are $(x_2, y_2)$, the distance between these two points is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Trigonometry Formulas

The six basic functions of Trigonometry are:

Pythagorean Identity

$\sin^2 \theta + \cos^2 \theta = 1$

Other Trigonometric Identities

• $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• $\sec \theta = \frac{1}{\cos \theta}$
• $\csc \theta = \frac{1}{\sin \theta}$