# Math Formulas

Maths is an abstract subject that needs a firm grasp of different Maths formulas. In-depth knowledge of Maths formulas prepares the students of Cass 6 to Class 12 to solve complex maths problems. Often students find it challenging to remember formulas and apply them in the right way. They only need to learn some tricks to memorize the formulas throughout their academic sessions.

## List of Maths Formulas

1. Algebra Formula
2. Algebraic Expressions Formula
3. Angle Formula
5. Area Formula
6. Area of a Pentagon Formula
7. Area of an Octagon Formula
8. Area Under the Curve Formula
9. Arithmetic Sequence Explicit Formula
10. Arithmetic Sequence Recursive Formula
11. Binary Formula
12. Calculus Formula
13. Change of Base Formula
14. Cofactor Formula
15. Complex Number Division Formula
16. Complex Number Power Formula
17. Conditional Probability Formula
18. Cos Inverse Formula
19. Cosine Formula
20. Cube Formula
21. Cubic Equation Formula
22. Daily Compound Interest Formula
23. Decay Formula
24. Decimal to Fraction Formula
25. Derivative Formula
26. Diagonal Formula
27. Equation Formula
28. Exponential Formula
29. Factorial Formula
30. Function Formulas
31. Geometric Distribution Formula
32. Graphs of Trigonometric Formula
33. HYPERBOLA FORMULA
34. Integral Formula
35. Lateral Area Formula
36. Limit Formula
37. LCM Formula
38. Newton’s Method Formula
39. Parabola Formula
40. Parallelogram Formula
41. Percentile Formula
42. Perimeter Formulas
43. Probability Distribution Formulas
44. Proportion Formula
45. Pythagorean Triples Formulas
47. Ratio Formula
48. Rhombus Formula
49. Sequence Formula
50. Sin Tan Formula
51. Sin Cos Formula
52. Surface Areas Formula
53. Statistics Formulas
54. Tangent Formula
55. Tan Theta Formula
56. Volume Charge Density Formula
57. The volume of a Cone Formula
58. The volume of a Cylinder Formula
59. The volume of a Rectangular Prism Formula
60. The volume of a Square Pyramid Formula
61. The volume of A Triangular Prism Formula
62. 30-60-90 Formulas
63. Vertex Formula
64. X and Y Intercept Formula
65. Z Score Formula

## BODMAS Formula

B = Bracket

O = Of

D = Division

M = Multiplication

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]

## All Math Formulas from Class 6 to Class 12

A detailed understanding of the Maths formulas makes the students of any standard perform better in examinations, whether it is class tests, final exams or board exams. Most of the chapters in the Maths syllabus are interrelated with each other. Therefore, if one understands the formulas of one chapter, the further chapters become simpler to them. Some of the interlinked chapters are percentage and profit-loss, percentages and fractions, real numbers and complex numbers, etc.

Students need to invest enough time and effort to analyse and understand the formulas methodically. The lists of Maths formulas are available for every chapter allocated in the latest syllabus of respective standards.

## Class 6 Maths formulas

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.

## Class 7 Maths formulas

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. 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

## Class 8 Math formulas

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

## Class 9 Math formulas

### 1. Real Numbers

1. √ab = √a √b
2. √(a/b) = √a / √b
3. (√a + √b) (√a – √b) = a – b
4. (√a + √b)2 = a + 2√ab + b
5. (a + √b) (a – √b) = a2 – b
6. (a + b) (a – b) = a2 – b2

### 2. Geometry Formulas  – Geometry Shapes Formulas for Class 9

Shape Formula Type Formula
Triangle Area

$\frac{1}{2} \times \text{base} \times \text{height}$

Perimeter

$a + b + c$

(where

$a$

,

$b$

, and

$c$

are the sides)

Heron’s Formula

$\sqrt{s(s-a)(s-b)(s-c)}$

, where

$s = \frac{a+b+c}{2}$

Rectangle Area

$\text{length} \times \text{breadth}$

Perimeter

$2 \times (\text{length} + \text{breadth})$

Square Area

$\text{side}^2$

Perimeter

$4 \times \text{side}$

Parallelogram Area

$\text{base} \times \text{height}$

Perimeter

$2 \times (\text{base} + \text{side})$

Rhombus Area

$\frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2$

Perimeter

$4 \times \text{side}$

Trapezium Area

$\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$

Perimeter

$\text{sum of all sides}$

Circle Area

$\pi \times \text{radius}^2$

Circumference

$2 \pi \times \text{radius}$

Cylinder Surface Area

$2 \pi \times \text{radius} \times (\text{height} + \text{radius})$

Volume

$\pi \times \text{radius}^2 \times \text{height}$

Cone Surface Area

$\pi \times \text{radius} \times (\text{radius} + \text{slant height})$

Volume

$\frac{1}{3} \pi \times \text{radius}^2 \times \text{height}$

Sphere Surface Area

$4 \pi \times \text{radius}^2$

Volume

$\frac{4}{3} \pi \times \text{radius}^3$

### 3. Algebra Identities for Class 9

Identity Formula
Square of a Sum

$(a + b)^2 = a^2 + 2ab + b^2$

Square of a Difference

$(a – b)^2 = a^2 – 2ab + b^2$

Product of a Sum and a Difference

$(a + b)(a – b) = a^2 – b^2$

Cube of a Sum

$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$

Cube of a Difference

$(a – b)^3 = a^3 – b^3 – 3ab(a – b)$

Sum of Cubes

$a^3 + b^3 = (a + b)(a^2 – ab + b^2)$

Difference of Cubes

$a^3 – b^3 = (a – b)(a^2 + ab + b^2)$

Expansion of

$(a + b + c)^2$

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

Expansion of

$(a + b + c)^3$

$(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)$

Square of a Binomial Sum

$(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$

Sum of Squares of Three Terms

$x^2 + y^2 + z^2 = (x + y + z)^2 – 2(xy + yz + zx)$

Square of a Binomial Difference

$(x – y – z)^2 = x^2 + y^2 + z^2 – 2xy + 2yz – 2zx$

### 4. Surface Area and Volume Formulas for Class 9

Here is a table with the surface area and volume formulas for various 3D shapes typically covered in Class 9:

Shape Formula Type Formula
Cuboid Surface Area

$2 (lb + bh + hl)$

Volume

$l \times b \times h$

Cube Surface Area

$6a^2$

Volume

$a^3$

Cylinder Curved Surface Area

$2 \pi r h$

Total Surface Area

$2 \pi r (r + h)$

Volume

$\pi r^2 h$

Cone Curved Surface Area

$\pi r l$

(where

$l$

is the slant height)

Total Surface Area

$\pi r (r + l)$

Volume

$\frac{1}{3} \pi r^2 h$

Sphere Surface Area

$4 \pi r^2$

Volume

$\frac{4}{3} \pi r^3$

Hemisphere Curved Surface Area

$2 \pi r^2$

Total Surface Area

$3 \pi r^2$

Volume

$\frac{2}{3} \pi r^3$

Prism Surface Area

$\text{Base Perimeter} \times h + 2 \times \text{Base Area}$

Volume

$\text{Base Area} \times h$

Pyramid Surface Area

$\text{Base Area} + \frac{1}{2} \times \text{Base Perimeter} \times l$

(where

$l$

is the slant height)

Volume

$\frac{1}{3} \times \text{Base Area} \times h$

These formulas are essential for solving various problems related to surface area and volume in Class 9.

### 5. Heron’s Formula

Here is a table of Heron’s Formula for calculating the area of a triangle, along with its components:

Formula Explanation

$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$

Heron’s Formula to calculate the area of a triangle
Components Explanation

$a, b, c$

The lengths of the sides of the triangle

$s = \frac{a + b + c}{2}$

The semi-perimeter of the triangle

This table encapsulates the essentials of Heron’s Formula used in Class 9 to find the area of a triangle when the lengths of all three sides are known.

### 6. Polynomial Formula

The general Polynomial Formula is,

F(x) = anxn + bxn-1 + an-2xn-2 + …….. + rx +s

### 7. Statistics (Measure of Central Tendency)

Here is a table summarizing the formulas for mean, median, and mode typically covered in Class 9:

Measure Formula
Mean

$\text{Mean} = \frac{\text{Sum of all observations}}{\text{Total number of observations}}$

Median For Odd Number of Observations:

$\left(\frac{n+1}{2}\right)^{th} \text{observation}$

For Even Number of Observations:

$\frac{\left(\frac{n}{2}\right)^{th} \text{observation} + \left(\left(\frac{n}{2}\right) + 1\right)^{th} \text{observation}}{2}$

Mode The value which occurs most frequently in the data

## Maths Formulas For Class 10

### 1. Algebra Formulas For Class 10

• (a+b)= a+ b+ 2ab
• (a-b)= a+ b– 2ab
• (a+b) (a-b) = a– 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)
• x+ y2 =½ [(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 = ½ [(x-y)2 + (y-z)2 + (z-x)2]

### 2. Basic Formulas For Powers For Class 10

• px p= pm+n
• {pm}⁄{pn} = pm-n
• (pm)= pmn
• p-m = 1/pm
• p1 = p
• P= 1

### 3. Arithmetic Formulas For Class 10

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

### 4. Trigonometry Formulas For Class 10

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. 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. Volume of Hemisphere = ⅔ x (πr3)
8. Lateral Surface Area of Hemisphere (LSA) = 2πr2
9. Total Surface Area of Hemisphere (TSA) = 3πr2
10. Volume of Prism = B × h
11. Lateral Surface Area of Prism (LSA) = p × h

## Class 11 Math formulas

### 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)

## Class 12 Math Formulas

### 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 for Class 6 to 12 for CBSE (NCERT) Board

If a student is searching for exciting ways to understand Maths formulas easily, Extramarks is the right place. The academic experts’ maths formula list helps the students of classes 6 to 12 understand the basics. The mentors carefully design the most significant Maths formulas list based on the chapters allotted to the current year syllabus. The following section presents the most vital formulas the students must study.

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