100 absolutely must know maths formulas for high schoolers

Here are 100 useful math formulas for high school students in India (ICSE/IB/CBSE) that cover various topics:

Algebra:

1. Quadratic formula: x = (-b ± √(b² – 4ac))/2a
2. Slope-intercept form: y = mx + b
3. Point-slope form: y – y1 = m(x – x1)
4. Distance formula: d = √((x2 – x1)² + (y2 – y1)²)
5. Midpoint formula: (x,y) = ((x1 + x2)/2, (y1 + y2)/2)
6. Standard form: Ax + By = C
7. Vertex form: y = a(x – h)² + k
8. FOIL method: (a + b)(c + d) = ac + ad + bc + bd
9. Difference of squares: (a + b)(a – b) = a² – b²
10. Perfect squares: (a + b)² = a² + 2ab + b²; (a – b)² = a² – 2ab + b²

Geometry:

11. Pythagorean theorem: a² + b² = c²
12. Area of a triangle: A = (1/2)bh
13. Area of a circle: A = πr²
14. Circumference of a circle: C = 2πr
15. Volume of a sphere: V = (4/3)πr³
16. Volume of a cylinder: V = πr²h
17. Volume of a cone: V = (1/3)πr²h
18. Perimeter of a rectangle: P = 2(l + w)
19. Perimeter of a square: P = 4s
20. Diagonal of a square: d = s√2

Trigonometry:

21. Sine: sinθ = opposite/hypotenuse
22. Cosine: cosθ = adjacent/hypotenuse
23. Tangent: tanθ = opposite/adjacent
24. Cosecant: cscθ = hypotenuse/opposite
25. Secant: secθ = hypotenuse/adjacent
26. Cotangent: cotθ = adjacent/opposite
27. Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ
28. Law of sines: a/sinA = b/sinB = c/sinC
29. Law of cosines: c² = a² + b² – 2abcosC
30. Unit circle: A circle with a radius of 1, used to determine trigonometric functions of angles.

Calculus:

31. Derivative of a constant: d/dx(c) = 0
32. Power rule: d/dx(x^n) = nx^(n-1)
33. Product rule: d/dx(fg) = fg’ + f’g
34. Quotient rule: d/dx(f/g) = (g f’ – f g’)/g^2
35. Chain rule: d/dx(f(g(x))) = f'(g(x))g'(x)
36. Integration by substitution: ∫f(g(x))g'(x)dx = ∫f(u)du, where u = g(x)
37. Integration by parts: ∫u dv = uv

Probability and Statistics:

38. Mean: The average of a set of numbers. It is calculated by summing all the numbers and then dividing by the total number of numbers.
39. Median: The middle value of a set of numbers when arranged in order.
40. Mode: The number that appears most frequently in a set of numbers.
41. Range: The difference between the largest and smallest values in a set of numbers.
42. Standard deviation: A measure of the spread of a set of numbers.
43. Variance: The average of the squared differences from the mean.
44. Probability of an event: P(A) = n(A)/n(S), where n(A) is the number of favorable outcomes and n(S) is the total number of outcomes.
45. Combinations: nCr = n!/r!(n-r)!
46. Permutations: nPr = n!/(n-r)!
47. Binomial distribution: P(x) = nCx p^x (1-p)^(n-x), where nCx = n!/x!(n-x)!
48. Normal distribution: A bell-shaped curve used to represent continuous data.

Number Theory:

49. Prime numbers: Numbers that are only divisible by 1 and themselves.
50. Composite numbers: Numbers that have more than two factors.
51. Factorization: Breaking down a number into its prime factors.
52. Greatest common factor (GCF): The largest factor that two or more numbers have in common.
53. Least common multiple (LCM): The smallest multiple that two or more numbers have in common.
54. Divisibility rules: Rules that help determine if a number is divisible by another number without performing division.
55. Euclid’s algorithm: A method for finding the GCF of two numbers.
56. Fermat’s little theorem: If p is a prime number, then a^p ≡ a (mod p) for any integer a.
57. Euler’s totient function: φ(n) is the number of positive integers less than or equal to n that are relatively prime to n.

Logic:

58. Proposition: A statement that is either true or false.
59. Conjunction: A compound statement formed by the word “and”.
60. Disjunction: A compound statement formed by the word “or”.
61. Negation: The opposite of a statement.
62. Conditional statement: A statement of the form “if p, then q”.
63. Contrapositive: The statement formed by switching the hypothesis and conclusion of a conditional statement and negating both.
64. Inverse: The statement formed by negating both the hypothesis and conclusion of a conditional statement.
65. Converse: The statement formed by switching the hypothesis and conclusion of a conditional statement.

Functions:

66. Domain: The set of all possible input values of a function.
67. Range: The set of all possible output values of a function.
68. One-to-one function: A function where each input has a unique output.
69. Inverse function: A function that “undoes” another function.
70. Composite function: A function that combines two or more functions.
71. Exponential function: A function of the form f(x) = a^x, where a is a positive constant.
72. Logarithmic function: A function of the form f(x) = loga(x), where a is a positive constant.
73. Natural logarithm: The logarithm with base e, where e is the mathematical constant approximately equal to 2.71828.

Matrices:

74. Matrix addition: C = A + B, where Cij = Aij + Bij for all i and j.
75. Matrix subtraction: C = A – B, where Cij = Aij – Bij for all i and j.
76. Scalar multiplication: C = kA, where Cij = kAij for all i and j.
77. Matrix multiplication: C = AB, where Cij = Σk AikBkj for all i and j.
78. Identity matrix: A square matrix with 1s on the diagonal and 0s everywhere else.
79. Transpose: A matrix where the rows and columns are interchanged.
80. Determinant: A scalar value that can be calculated for a square matrix.
81. Inverse: The matrix that when multiplied by a given matrix gives the identity matrix.
82. Adjoint: The transpose of the matrix of cofactors.
83. Cofactor: The product of (-1)^(i+j) and the determinant of the matrix obtained by deleting the ith row and jth column of a given matrix.
84. Geometry:
85. Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the lengths of the two legs of a right triangle and c is the length of the hypotenuse.
86. Trigonometric ratios: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent.
87. Similar triangles: Triangles with the same shape but different sizes.
88. Congruent triangles: Triangles with the same shape and size.
89. Area of a triangle: 1/2 bh, where b is the base and h is the height.
90. Area of a circle: πr^2, where r is the radius.
91. Circumference of a circle: 2πr, where r is the radius.
92. Volume of a sphere: 4/3 πr^3, where r is the radius.
93. Surface area of a sphere: 4πr^2, where r is the radius.
94. Midpoint formula: (x1 + x2)/2, (y1 + y2)/2.
95. Distance formula: √(x2-x1)^2 + (y2-y1)^2.
96. Calculus:
97. Derivative: The slope of a tangent line to a curve at a given point.
98. Chain rule: d/dx (f(g(x))) = f'(g(x))g'(x).
99. Product rule: (fg)’ = f’g + fg’.
100. Quotient rule: (f/g)’ = (f’g – fg’)/g^2.
101. Integration by substitution: ∫f(g(x))g'(x)dx = ∫f(u)du.
102. Fundamental theorem of calculus: ∫a^b f(x)dx = F(b) – F(a), where F(x) is the antiderivative of f(x)