Maths Formulas: Solved Examples and Essential Maths Formulas for Every Student | Class 6 to 12

Maths formulas: Get an Overview

Maths formulas are the building blocks for solving complex problems and unlocking the secrets of the universe. Whether you're a student, a professional in a STEM field, or simply someone who wants to sharpen their analytical skills, understanding and applying mathematical formulas is essential. In this article, we will delve into the world of mathematical formulas, exploring their significance, and providing a comprehensive guide to help you master them. Let's embark on this exciting journey of mathematical discovery!

A mathematical formula is a precise representation or rule that emerges from the relationship between multiple quantities, and its outcome is expressed using symbolic notation. Math formulas consist of various elements, such as constants, which are specific numbers, variables that represent unknown values, mathematical signs and symbols, and occasionally exponential powers.

BODMAS Formula

B = Bracket 

O = Of

D = Division

M = Multiplication 

A = Addition

S = Subtraction 

Basic Algebra Formulas

1. a2 – b2 = (a – b)(a + b)
2. (a + b)2 = a2 + 2ab + b2
3. a2 + b2 = (a + b)2 – 2ab
4. (a – b)2 = a2 – 2ab + b2
5. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
6. (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
7. (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
8. (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
9. a3 – b3 = (a – b)(a2 + ab + b2)
10. a3 + b3 = (a + b)(a2 – ab + b2)
11. (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
12. (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
13. a4 – b4 = (a – b)(a + b)(a2 + b2)
14. a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
15. (a +b+ c)2=a2+b2+c2+2ab+2bc+2ca
16. (a +b+ c+…)2=a2+b2+c2+⋯+2(ab +ac+ bc +⋯
17. (x+ y+ z)2=x2+y2+z2+2xy+2yz+2xz
18. (x +y−z)2=x2+y2+z2+2xy−2yz−2xz
19. (x− y+ z)2=x2+y2+z2−2xy−2yz+2xz
20. (x−y−z)2=x2+y2+z2−2xy+2yz−2xz
21. x3+y3+z3−3xyz=(x+ y+ z)(x2+y2+z2−xy−yz−xz)
22. x2+y2=1/2[(x+ y)2+(x−y)2]
23. (x +a)(x +b)(x +c)=x3+(a +b+ c)x2+(ab +bc+ ca)x+ abc
24. x3+y3=(x+ y)(x2−xy+y2)
25. x3−y3=(x−y)(x2+xy+y2)
26. x2+y2+z2−xy−yz−zx=1/2[(x−y)2+(y−z)2+(z−x)2]

Maths Formulas Table

The table below has all the basic maths formulas:

Maths formulas: Important Maths Formulas from Class 6-12

Gaining a comprehensive understanding of mathematics formulas significantly enhances students' performance in examinations across all academic levels, be it class tests, final exams, or board exams. Most of the chapters within the mathematics syllabus are interconnected, meaning that mastering the formulas of one chapter simplifies comprehension of subsequent chapters. Noteworthy examples of interrelated chapters include percentage and profit-loss, percentages and fractions, real numbers and complex numbers, among others.

To truly grasp the formulas, students must allocate sufficient time and effort to systematically analyze and comprehend them. Detailed lists of math formulas are readily available for each chapter as per the latest syllabus of respective academic standards.

Maths Formulas: Class 6

1. 1,000,000,000 is called one billion.
2. Anything divided by zero is called ‘undefined'.
3. A number is divisible by 2 if it has 0, 2, 4, 6 or 8 in one place.
4. A number is divisible by 3 if the sum of the digits is a multiple of 3.
5. A simple closed figure formed by line segments is a polygon. Triangle is a three-sided Polygon. Quadrilaterals are four-sided polygons.
6. An equation is a condition represented on a variable. An equation is composed of two sides, known as the Left-Hand Side and Right Hand Side, separated by an equal (=) sign. 
7. The perimeter of a Square = 4 × Length of its side
8. Perimeter of a Rectangle = 2 × (Length + Breadth)
9. The perimeter of an Equilateral triangle = 3 × Length of a side
10. Area of a Rectangle = length × breadth
11. Variable refers to a value that is not fixed. It can take different values.
12. An equation is a condition represented on a variable.
13. An equation is composed of two sides, known as the Left-Hand Side and Right Hand Side, separated by an equal (=) sign.

Maths Formulas: Class 7

1. Product of rational numbers = (Product of Numerators) / (Product of Denominators)
2. First Rational Number × (Reciprocal of other Rational Number)
3. Area of a Square = Side2
4. The perimeter of a Square = 4 × Side
5. Area of Rectangle = Length × Breadth
6. Perimeter of a Rectangle = 2 × (Length + Breadth)
7. Area of a Parallelogram = Base × Height
8. Area of Triangle = 1/ 2 × Base × Height
9. Circumference of a circle = π d, where ‘d' is the diameter of a circle and π = 22/7 or 3.14
10. Area of a circle = πr2
11. Law of Product: am × an = am+n
12. Law of Quotient: am/an = am-n
13. Law of Zero Exponent: a0 = 1
14. Law of Negative Exponent: a-m = 1/am
15. Law of Power of a Power: (am)n = amn
16. Law of Power of a Product: (ab)m = ambm
17. Law of Power of a Quotient: (a/b)m = am/bm
18. (a-b) 2 = a2 – 2ab + b2 
19. (a-b-c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ac

If their two ratios are equivalent for any four quantities, those four quantities are said to be proportionate.

Increase in Percentage = (Change / Original Amount) × 100

Profit Percentage = (Profit / Cost price) × 100

Simple Interest = (Principal × Rate × Time) / 100

Amount = Principal + Interest

Read more about the Straight Line Formula.

Maths Formulas: Class 8

1. Additive inverse of rational number: a/b = -b/a
2. Multiplicative Inverse of a/b = c/d , if a/b × c/d = 1
3. Distributives a(b – c) = ab – ac
4. Probability of the occurrence of an event = Number of outcomes that comprise an event/ Total number of outcomes
5. Compound Interest formula = Amount – Principal, Amount in case the interest is calculated annually = Principal (1 + Rate/100)n, where ‘n' is the period.
6. (a – b)2 = a2 – 2ab + b2
7. (a + b) (a – b) = a2 – b2  
8. Euler's Formula: For any polyhedron, Number of faces + Number of vertices – Number of edges = 2
9. Volume of a Cone = (1 / 3)πr2h
10. Volume of a Sphere = (4/3) π r3

Read More About: 

Maths Formulas: Class 9

Read more about the Collinear Points.

Maths Formulas: Class 10

Algebra Formulas

1. (a + b)2 = a2 + 2ab + b2
2. (a – b)2 = a2 – 2ab + b2
3. (a + b) (a – b) = a2 – b2
4. (x + a)(x + b) = x2 + (a + b)x + ab
5. (x + a)(x – b) = x2 + (a – b)x – ab
6. (x – a)(x + b) = x2 + (b – a)x – ab
7. (x – a)(x – b) = x2 – (a + b)x + ab
8. (a + b)3 = a3 + b3 + 3ab(a + b)
9. (a – b)3 = a3 – b3 – 3ab(a – b)
10. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
11. (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
12. (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
13. (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
14. x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – xz)

Arithmetic Formulas 

1. an = a + (n – 1) d, where an is the nth term.
2. Sn= n/2 [2a + (n – 1)d]

Trigonometry Formulas

1. sin(90° – A) = cos A
2. cos(90° – A) = sin A
3. tan(90° – A) = cot A
4. cot(90° – A) = tan A
5. sec(90° – A) = cosec A
6. cosec(90° – A) = sec A
7. sin2 θ + cos2 θ = 1 ⇒sin2 θ = 1 – cos2 θ ⇒cos2 θ = 1 – sin2 θ
8. cosec2 θ – cot2 θ = 1 ⇒cosec2 θ = 1 + cot2 θ ⇒cot2 θ = cosec2 θ – 1
9. sec2 θ – tan2 θ = 1 ⇒sec2 θ = 1 + tan2 θ ⇒tan2 θ = sec2 θ – 1
10. sin θ cosec θ = 1 ⇒cos θ sec θ = 1 ⇒tan θ cot θ = 1

Circle Formula

1. The tangent to a circle equation x2 + y2 = a2 for a line y = mx + c is given by the equation y = mx ± a √ [1+ m2].
2. The tangent to a circle equation x2 + y2 = a2 at (a1,b1) is xa1 + yb1 = a2

Area and Volume Formulas 

1. The volume of Sphere = 4/3 ×π r3
2. Lateral Surface Area of Sphere (LSA) = 4π r2
3. Total Surface Area of Sphere (TSA) = 4πr2
4. The volume of the Right Circular Cylinder = πr2h
5. Lateral Surface Area of Right Circular Cylinder (LSA) = 2×(πrh)
6. Total Surface Area of Right Circular Cylinder (TSA) = 2πr×(r + h)
7. The volume of Hemisphere = ⅔ x (πr3)
8. Lateral Surface Area of Hemisphere (LSA) = 2πr2
9. Total Surface Area of Hemisphere (TSA) = 3πr2
10. The volume of Prism = B × h
11. Lateral Surface Area of Prism (LSA) = p × h

Maths formulas: Class 11

Algebra Formulas

1. a × (b + c) = a × b + a × c (Distributive property)
2. a + b = b + a (Commutative Property of Addition)
3. a × b = b × a (Commutative Property of Multiplication)
4. a + (b + c) = (a + b) + c (Associative Property of Addition)
5. a × (b × c) = (a × b) × c (Associative Property of Multiplication)
6. a + 0 = a (Additive Identity Property)
7. a × 1 = a(Multiplicative Identity Property)
8. a + (-a) = 0 (Additive Inverse Property)
9. a⋅(1/a) = 1 (Multiplicative Inverse Property)
10. a × (0) =0 (Zero Property of Multiplication)

Trigonometry Formulas

1. sin(90° – A) = cos A
2. cos(90° – A) = sin A
3. tan(90° – A) = cot A
4. cot(90° – A) = tan A
5. sec(90° – A) = cosec A
6. cosec(90° – A) = sec A
7. sin2 θ + cos2 θ = 1 ⇒sin2 θ = 1 – cos2 θ ⇒cos2 θ = 1 – sin2 θ
8. cosec2 θ – cot2 θ = 1 ⇒cosec2 θ = 1 + cot2 θ ⇒cot2 θ = cosec2 θ – 1
9. sec2 θ – tan2 θ = 1 ⇒sec2 θ = 1 + tan2 θ ⇒tan2 θ = sec2 θ – 1
10. sin θ cosec θ = 1 ⇒cos θ sec θ = 1 ⇒tan θ cot θ = 1

Calculus Formulas

1. d/dx [f(x) + g (x)] = d/dx [f(x)] + d/dx [g(x)]
2. d/dx [f(x) – g (x)] = d/dx [f(x)] – d/dx [g(x)]
3. d/dx [f(x) × g (x)] = d/dx [f(x)] × [g(x)] + [f(x)] × d/dx [g(x)]
4. d/dx [f(x) / g (x)] = {d/dx [f(x)] × [g(x)] – [f(x)] × d/dx [g(x)]} / g(x)2

Geometry and Lines Formulas 

1. Slope m = rise/run = Δy/Δx = y2−y1/x2−x1
2. Point-Slope Form y−y1 = m (x−x1)

Maths Formulas: Class12

Vector Formulas

1. A + B = B + A (Commutative Law)
2. A + (B + C) = (A + B) + C (Associative Law)
3. (A • B )= |P| |Q| cos θ ( Dot Product )
4. (A × B )= |P| |Q| sin θ (Cross Product)
5. k (A + B )= kA + kB
6. A + 0 = 0 + A (Additive Identity)

Trigonometry Formulas

1. sin-1(-x) = – sin-1x 
2. tan-1x + cot-1x = π / 2
3. sin-1x + cos-1 x = π / 2
4. cos-1(-x) = π – cos-1x
5. cot-1(-x) = π – cot-1x

Calculus Formulas

1. ∫ f(x) dx = F(x) + C
2. Power Rule: ∫ xn dx = (xn+1) / (n+1) + C. (Where n ≠ -1)
3. Exponential Rules: ∫ ex dx = ex + C 
4. ∫ ax dx = ax / ln(a) + C
5. ∫ ln(x) dx = x ln(x) – x + C
6. Constant Multiplication Rule: ∫ a dx = ax + C, where a is the constant. 
7. Reciprocal Rule: ∫ (1/x) dx = ln(x)+ C 
8. Sum Rules: ∫ [f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
9. Difference Rules: ∫ [f(x) – g(x)] dx = ∫f(x) dx – ∫g(x) dx
10. ∫k f(x) dx = k ∫f(x) dx, , where k is any real number.
11. Integration by parts: ∫ f(x) g(x) dx = f(x) ∫ g(x) dx – ∫[d/dx f(x) × ∫ g(x) dx]dx
12. ∫cos x dx = sin x + C
13. ∫ sin x dx = -cos x + C
14. ∫ sec2 x dx = tan x + C
15. ∫ cosec2 x dx = -cot x + C
16. ∫ sec x tan x dx = sec x + C
17. ∫ cosec x cot x dx = – cosec x + C

Geometry Formulas 

1. Cartesian equation of a plane: lx + my + nz = d 
2. Distance between two points P(x1, y1, z1) and Q(x2, y2, z2): PQ = √ ((x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2) 

Maths Formulas: Conclusion

Mathematical formulas serve as powerful tools for solving problems across various disciplines. Understanding and mastering these formulas can significantly enhance your problem-solving skills and analytical thinking. From algebraic equations to geometric measurements, trigonometric identities to calculus operations, and probability to statistics, the world of mathematical formulas is vast and interconnected. By familiarizing yourself with these formulas, you gain the confidence to tackle complex mathematical challenges and unlock new realms of knowledge. So, embrace the power of mathematics and let the formulas guide you on your journey of exploration and success.