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Home > Articles > Maths Formulas: Solved Examples and Essential Maths Formulas for Every Student | Class 6 to 12
Maths Formulas: Solved Examples and Essential Maths Formulas for Every Student | Class 6 to 12
Kasturi Talukdar
Updated on 18th September, 2024 , 10 min read
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
- a2 – b2 = (a – b)(a + b)
- (a + b)2 = a2 + 2ab + b2
- a2 + b2 = (a + b)2 – 2ab
- (a – b)2 = a2 – 2ab + b2
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
- (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
- (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
- a3 – b3 = (a – b)(a2 + ab + b2)
- a3 + b3 = (a + b)(a2 – ab + b2)
- (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
- (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
- a4 – b4 = (a – b)(a + b)(a2 + b2)
- a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
- (a +b+ c)2=a2+b2+c2+2ab+2bc+2ca
- (a +b+ c+…)2=a2+b2+c2+⋯+2(ab +ac+ bc +⋯
- (x+ y+ z)2=x2+y2+z2+2xy+2yz+2xz
- (x +y−z)2=x2+y2+z2+2xy−2yz−2xz
- (x− y+ z)2=x2+y2+z2−2xy−2yz+2xz
- (x−y−z)2=x2+y2+z2−2xy+2yz−2xz
- x3+y3+z3−3xyz=(x+ y+ z)(x2+y2+z2−xy−yz−xz)
- x2+y2=1/2[(x+ y)2+(x−y)2]
- (x +a)(x +b)(x +c)=x3+(a +b+ c)x2+(ab +bc+ ca)x+ abc
- x3+y3=(x+ y)(x2−xy+y2)
- x3−y3=(x−y)(x2+xy+y2)
- 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:
Perimeter |
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Circumference |
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Area |
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Surface Area |
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Volume |
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Pythagoras Theorem | a2 + b2 = c2 | |
Distance Formula | d = √[(x2 – x1)2 +(y2 – y1)2] | |
Slope of a line | m = y2 – y1 / x2 – x1 | |
Mid- Point Formula | M = [(x1 + x2 )/ 2 , (y1 + y2 )/ 2] | |
Algebraic Formula |
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Trigonometric Formulas |
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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
- 1,000,000,000 is called one billion.
- Anything divided by zero is called ‘undefined'.
- A number is divisible by 2 if it has 0, 2, 4, 6 or 8 in one place.
- A number is divisible by 3 if the sum of the digits is a multiple of 3.
- A simple closed figure formed by line segments is a polygon. Triangle is a three-sided Polygon. Quadrilaterals are four-sided polygons.
- 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.
- The perimeter of a Square = 4 × Length of its side
- Perimeter of a Rectangle = 2 × (Length + Breadth)
- The perimeter of an Equilateral triangle = 3 × Length of a side
- Area of a Rectangle = length × breadth
- Variable refers to a value that is not fixed. It can take different values.
- 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.
Maths Formulas: Class 7
- Product of rational numbers = (Product of Numerators) / (Product of Denominators)
- First Rational Number × (Reciprocal of other Rational Number)
- Area of a Square = Side2
- The perimeter of a Square = 4 × Side
- Area of Rectangle = Length × Breadth
- Perimeter of a Rectangle = 2 × (Length + Breadth)
- Area of a Parallelogram = Base × Height
- Area of Triangle = 1/ 2 × Base × Height
- Circumference of a circle = π d, where ‘d' is the diameter of a circle and π = 22/7 or 3.14
- Area of a circle = πr2
- Law of Product: am × an = am+n
- Law of Quotient: am/an = am-n
- Law of Zero Exponent: a0 = 1
- Law of Negative Exponent: a-m = 1/am
- Law of Power of a Power: (am)n = amn
- Law of Power of a Product: (ab)m = ambm
- Law of Power of a Quotient: (a/b)m = am/bm
- (a-b) 2 = a2 – 2ab + b2
- (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
- Additive inverse of rational number: a/b = -b/a
- Multiplicative Inverse of a/b = c/d , if a/b × c/d = 1
- Distributives a(b – c) = ab – ac
- Probability of the occurrence of an event = Number of outcomes that comprise an event/ Total number of outcomes
- Compound Interest formula = Amount – Principal, Amount in case the interest is calculated annually = Principal (1 + Rate/100)n, where ‘n' is the period.
- (a – b)2 = a2 – 2ab + b2
- (a + b) (a – b) = a2 – b2
- Euler's Formula: For any polyhedron, Number of faces + Number of vertices – Number of edges = 2
- Volume of a Cone = (1 / 3)πr2h
- Volume of a Sphere = (4/3) π r3
Read More About:
Maths Formulas: Class 9
Topic | Shapes/Statistics | Maths Formulas |
Real Numbers | - |
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Geometry Formulas | Rectangle | A = Length x Width P = 2(Length + Width) |
Triangle | A = ½ x Breadth x Height P = Sum of all the three sides of a triangle | |
Trapezoid | A = ½ x Height x (b₁x b₂) P = Sum of all the sides of a trapezoid | |
Parallelogram | A = Breadth x Height P = 2( a+ b) Here. a = side B = base | |
Circle | A = πr² P = 2 πr | |
Algebra Identities | - |
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Surface Area and Volume | Cuboid | 2 = (lb + bh + hl), Here, l = length, b = Breadth, h = height V = Length x Breadth x Height |
Cube | A = 6 side² V = Side³ Cylinder A = 2πr( h + r) Here, r = radius of circular cylinder H = height of a cylinder V = πr²H | |
Cone | A = πr( L + r) Here, l = slant height r = Radius of base Also, l² = h² + r², where h is the cone's height V = ½ πr² | |
Sphere | A = 4πr² V = 4/3πr³ | |
Heron's Formula | - | Area of Triangle with 3 sides √s(s-a)(s-b)(s-c) Here, s = semi perimeter A,b, c are the sides of a triangle. Semi Perimeter S = ( a + b + c)/2 |
Polynomial Formula | - | P (x) = anxn + an- 1xn- 1 – an- 2xn- 1 + …… ax + a0 |
Statistics (Measure of Central Tendency) | Mean | Sum of all the observations/ Total Number of Observations |
Median | For odd observations = ((n+1)/2)th observations For even Observations – ((n/2)th + ((n/2) +1)th)/2 observations | |
Mode | The value which occurs most frequently in a data |
Read more about the Collinear Points.
| FAQs on Maths Formulas |
| Q. What are the basic formulas of maths? |
A. The following formulas are very popular amongst Elementary & Middle School students:
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| Q. What is the most important math formula? |
| A. Pythagoras' Theorem has contributed significantly to the growth and development of mathematics as a discipline. This theorem is widely applied in geometry and trigonometry, essential branches of mathematics. |
| Q. What is the famous mathematical formula? |
| A. Pythagorean Theorem (c² = a² + b²) is very popular and widely studied in mathematics. |
| Q. What is basic math? |
| A. Basic math involves the following: addition, subtraction, multiplication, and division. Some basic fundamental mathematical concepts include learning shapes, patterns, fractions, decimals, percentages, exponents, ratios, scientific notation, and formulas. |
| Q. What is Ramanujan formulas? |
| A. In number theory, a branch of mathematics, the Ramanujan-Nagell equation is an equation between a square number and a number seven less than a power of two: it is an exponential Diophantine equation, that is to say an equation to be solved in integers where one of the variables appears as an exponent. |
Maths Formulas: Class 10
Algebra Formulas
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- (a + b) (a – b) = a2 – b2
- (x + a)(x + b) = x2 + (a + b)x + ab
- (x + a)(x – b) = x2 + (a – b)x – ab
- (x – a)(x + b) = x2 + (b – a)x – ab
- (x – a)(x – b) = x2 – (a + b)x + ab
- (a + b)3 = a3 + b3 + 3ab(a + b)
- (a – b)3 = a3 – b3 – 3ab(a – b)
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
- (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
- (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
- (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
- x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – xz)
Arithmetic Formulas
- an = a + (n – 1) d, where an is the nth term.
- Sn= n/2 [2a + (n – 1)d]
Trigonometry Formulas
- sin(90° – A) = cos A
- cos(90° – A) = sin A
- tan(90° – A) = cot A
- cot(90° – A) = tan A
- sec(90° – A) = cosec A
- cosec(90° – A) = sec A
- sin2 θ + cos2 θ = 1 ⇒sin2 θ = 1 – cos2 θ ⇒cos2 θ = 1 – sin2 θ
- cosec2 θ – cot2 θ = 1 ⇒cosec2 θ = 1 + cot2 θ ⇒cot2 θ = cosec2 θ – 1
- sec2 θ – tan2 θ = 1 ⇒sec2 θ = 1 + tan2 θ ⇒tan2 θ = sec2 θ – 1
- sin θ cosec θ = 1 ⇒cos θ sec θ = 1 ⇒tan θ cot θ = 1
Circle Formula
- 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].
- The tangent to a circle equation x2 + y2 = a2 at (a1,b1) is xa1 + yb1 = a2
Area and Volume Formulas
- The volume of Sphere = 4/3 ×π r3
- Lateral Surface Area of Sphere (LSA) = 4π r2
- Total Surface Area of Sphere (TSA) = 4πr2
- The volume of the Right Circular Cylinder = πr2h
- Lateral Surface Area of Right Circular Cylinder (LSA) = 2×(πrh)
- Total Surface Area of Right Circular Cylinder (TSA) = 2πr×(r + h)
- The volume of Hemisphere = ⅔ x (πr3)
- Lateral Surface Area of Hemisphere (LSA) = 2πr2
- Total Surface Area of Hemisphere (TSA) = 3πr2
- The volume of Prism = B × h
- Lateral Surface Area of Prism (LSA) = p × h
Maths formulas: Class 11
Algebra Formulas
- a × (b + c) = a × b + a × c (Distributive property)
- a + b = b + a (Commutative Property of Addition)
- a × b = b × a (Commutative Property of Multiplication)
- a + (b + c) = (a + b) + c (Associative Property of Addition)
- a × (b × c) = (a × b) × c (Associative Property of Multiplication)
- a + 0 = a (Additive Identity Property)
- a × 1 = a(Multiplicative Identity Property)
- a + (-a) = 0 (Additive Inverse Property)
- a⋅(1/a) = 1 (Multiplicative Inverse Property)
- a × (0) =0 (Zero Property of Multiplication)
Trigonometry Formulas
- sin(90° – A) = cos A
- cos(90° – A) = sin A
- tan(90° – A) = cot A
- cot(90° – A) = tan A
- sec(90° – A) = cosec A
- cosec(90° – A) = sec A
- sin2 θ + cos2 θ = 1 ⇒sin2 θ = 1 – cos2 θ ⇒cos2 θ = 1 – sin2 θ
- cosec2 θ – cot2 θ = 1 ⇒cosec2 θ = 1 + cot2 θ ⇒cot2 θ = cosec2 θ – 1
- sec2 θ – tan2 θ = 1 ⇒sec2 θ = 1 + tan2 θ ⇒tan2 θ = sec2 θ – 1
- sin θ cosec θ = 1 ⇒cos θ sec θ = 1 ⇒tan θ cot θ = 1
Calculus Formulas
- d/dx [f(x) + g (x)] = d/dx [f(x)] + d/dx [g(x)]
- d/dx [f(x) – g (x)] = d/dx [f(x)] – d/dx [g(x)]
- d/dx [f(x) × g (x)] = d/dx [f(x)] × [g(x)] + [f(x)] × d/dx [g(x)]
- 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
- Slope m = rise/run = Δy/Δx = y2−y1/x2−x1
- Point-Slope Form y−y1 = m (x−x1)
Maths Formulas: Class12
Vector Formulas
- A + B = B + A (Commutative Law)
- A + (B + C) = (A + B) + C (Associative Law)
- (A • B )= |P| |Q| cos θ ( Dot Product )
- (A × B )= |P| |Q| sin θ (Cross Product)
- k (A + B )= kA + kB
- A + 0 = 0 + A (Additive Identity)
Trigonometry Formulas
- sin-1(-x) = – sin-1x
- tan-1x + cot-1x = π / 2
- sin-1x + cos-1 x = π / 2
- cos-1(-x) = π – cos-1x
- cot-1(-x) = π – cot-1x
Calculus Formulas
- ∫ f(x) dx = F(x) + C
- Power Rule: ∫ xn dx = (xn+1) / (n+1) + C. (Where n ≠ -1)
- Exponential Rules: ∫ ex dx = ex + C
- ∫ ax dx = ax / ln(a) + C
- ∫ ln(x) dx = x ln(x) – x + C
- Constant Multiplication Rule: ∫ a dx = ax + C, where a is the constant.
- Reciprocal Rule: ∫ (1/x) dx = ln(x)+ C
- Sum Rules: ∫ [f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
- Difference Rules: ∫ [f(x) – g(x)] dx = ∫f(x) dx – ∫g(x) dx
- ∫k f(x) dx = k ∫f(x) dx, , where k is any real number.
- Integration by parts: ∫ f(x) g(x) dx = f(x) ∫ g(x) dx – ∫[d/dx f(x) × ∫ g(x) dx]dx
- ∫cos x dx = sin x + C
- ∫ sin x dx = -cos x + C
- ∫ sec2 x dx = tan x + C
- ∫ cosec2 x dx = -cot x + C
- ∫ sec x tan x dx = sec x + C
- ∫ cosec x cot x dx = – cosec x + C
Geometry Formulas
- Cartesian equation of a plane: lx + my + nz = d
- 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.
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Frequently Asked Questions
What are some commonly used math formulas?
Some commonly used math formulas include the Pythagorean theorem, quadratic formula, area and perimeter formulas for geometric shapes, trigonometric identities, and formulas for arithmetic and geometric sequences.
How can I remember math formulas more easily?
To remember math formulas, practice applying them in various problems, create flashcards or cheat sheets, break formulas into smaller parts for better understanding, and try to understand the logic behind each formula.
Where can I find a comprehensive list of math formulas?
You can find comprehensive lists of math formulas in textbooks, educational websites, math reference books, or by searching online math resources.
Are there any shortcuts or tricks for learning math formulas?
Yes, some tips include identifying patterns in formulas, understanding the derivations or proofs of formulas, and connecting formulas to real-world applications to enhance understanding and memory.
How can I use math formulas effectively in problem-solving?
To use math formulas effectively, read the problem carefully, identify the relevant formula(s) needed, substitute the given values into the formula, and solve step-by-step using proper mathematical operations.
Can I derive math formulas by myself?
Yes, some math formulas can be derived using logical reasoning, algebraic manipulations, or geometric principles. Deriving formulas can deepen your understanding of the underlying concepts.
What are the essential math formulas for geometry?
Essential geometry formulas include those for area and perimeter of shapes (rectangle, triangle, circle), volume and surface area of solids (cuboid, cylinder, sphere), and trigonometric ratios for angles and triangles.
Are there any formulas to calculate statistical measures?
Yes, statistical measures like mean, median, mode, and standard deviation have specific formulas. For example, mean is calculated by dividing the sum of all observations by the total number of observations.
Can I use calculators to find math formulas?
Calculators can help in performing calculations involved in math formulas, but understanding the concepts behind the formulas and knowing when and how to apply them is crucial.