Simple and useful math formulas

🔢 Basic Arithmetic

1. a+b=b+aa + b = b + a (Commutative Property of Addition)
2. a×b=b×aa \times b = b \times a (Commutative Property of Multiplication)
3. (a+b)+c=a+(b+c)(a + b) + c = a + (b + c) (Associative Property of Addition)
4. (a×b)×c=a×(b×c)(a \times b) \times c = a \times (b \times c) (Associative Property of Multiplication)
5. a(b+c)=ab+aca(b + c) = ab + ac (Distributive Property)
6. a−(−b)=a+ba – (-b) = a + b
7. a÷b=a×1ba \div b = a \times \frac{1}{b}
8. a0=1a^0 = 1 (any number except 0)
9. a1=aa^1 = a
10. a−n=1ana^{-n} = \frac{1}{a^n}

✖️ Multiplication Tables (Quick Formulae)

11. n×0=0n \times 0 = 0
12. n×1=nn \times 1 = n
13. n×10=n0n \times 10 = n0
14. n×11=n0+nn \times 11 = n0 + n
15. 2n=n+n2n = n + n

🧮 Algebra

16. (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
17. (a−b)2=a2−2ab+b2(a – b)^2 = a^2 – 2ab + b^2
18. (a+b)(a−b)=a2−b2(a + b)(a – b) = a^2 – b^2
19. a2−b2=(a+b)(a−b)a^2 – b^2 = (a + b)(a – b)
20. (x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a + b)x + ab
21. ax+b=0⇒x=−baax + b = 0 \Rightarrow x = -\frac{b}{a}
22. ax2+bx+c=0⇒x=−b±b2−4ac2aax^2 + bx + c = 0 \Rightarrow x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
23. (x+y)3=x3+3x2y+3xy2+y3(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
24. (x−y)3=x3−3x2y+3xy2−y3(x – y)^3 = x^3 – 3x^2y + 3xy^2 – y^3
25. x3−y3=(x−y)(x2+xy+y2)x^3 – y^3 = (x – y)(x^2 + xy + y^2)

📐 Geometry

2D Shapes:

26. Perimeter of square = 4a4a
27. Area of square = a2a^2
28. Perimeter of rectangle = 2(l+b)2(l + b)
29. Area of rectangle = l×bl \times b
30. Area of triangle = 12×b×h\frac{1}{2} \times b \times h
31. Area of circle = πr2\pi r^2
32. Circumference = 2πr2\pi r
33. Diameter = 2r2r
34. Area of parallelogram = b×hb \times h
35. Area of trapezium = 12(a+b)h\frac{1}{2}(a + b)h

3D Shapes:

36. Volume of cube = a3a^3
37. Volume of cuboid = l×b×hl \times b \times h
38. Volume of sphere = 43πr3\frac{4}{3}\pi r^3
39. Volume of cylinder = πr2h\pi r^2 h
40. Volume of cone = 13πr2h\frac{1}{3}\pi r^2 h
41. Surface area of cube = 6a26a^2
42. Surface area of cuboid = 2(lb+bh+hl)2(lb + bh + hl)

📏 Mensuration

43. Diagonal of rectangle = l2+b2\sqrt{l^2 + b^2}
44. Diagonal of square = 2a\sqrt{2}a
45. Height of equilateral triangle = 32a\frac{\sqrt{3}}{2}a
46. Radius from area of circle = Aπ\sqrt{\frac{A}{\pi}}
47. Slant height of cone = r2+h2\sqrt{r^2 + h^2}

📉 Trigonometry (Basic)

48. sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
49. tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}
50. sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}
51. csc⁡x=1sin⁡x\csc x = \frac{1}{\sin x}
52. cot⁡x=1tan⁡x\cot x = \frac{1}{\tan x}
53. sin⁡(2x)=2sin⁡xcos⁡x\sin(2x) = 2\sin x\cos x
54. cos⁡(2x)=cos⁡2x−sin⁡2x\cos(2x) = \cos^2 x – \sin^2 x
55. tan⁡(2x)=2tan⁡x1−tan⁡2x\tan(2x) = \frac{2\tan x}{1 – \tan^2 x}

🧭 Coordinate Geometry

56. Distance formula = (x2−x1)2+(y2−y1)2\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}
57. Midpoint = (x1+x22,y1+y22)\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
58. Slope = y2−y1x2−x1\frac{y_2 – y_1}{x_2 – x_1}
59. Equation of line = y=mx+cy = mx + c
60. General line: Ax+By+C=0Ax + By + C = 0

📊 Statistics

61. Mean = Sum of observationsNumber of observations\frac{\text{Sum of observations}}{\text{Number of observations}}
62. Median = Middle value in sorted list
63. Mode = Most frequent value
64. Range = Max – Min
65. Variance = ∑(xi−xˉ)2n\frac{\sum(x_i – \bar{x})^2}{n}
66. Standard deviation = variance\sqrt{\text{variance}}
67. Mean of first nn natural numbers = n(n+1)2n\frac{n(n+1)}{2n}
68. Mean of squares = n(n+1)(2n+1)6n\frac{n(n+1)(2n+1)}{6n}

🎲 Probability

69. Probability = Favorable outcomesTotal outcomes\frac{\text{Favorable outcomes}}{\text{Total outcomes}}
70. P(E)+P(not E)=1P(E) + P(\text{not E}) = 1
71. Compound Probability (Independent): P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)
72. Union (Mutually Exclusive): P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)

🔢 Number System

73. Even Number = 2n2n
74. Odd Number = 2n+12n + 1
75. Prime Number: Only divisible by 1 and itself
76. LCM of (a,b) = abGCD(a,b)\frac{ab}{\text{GCD}(a,b)}
77. HCF = Highest common factor

⏱ Time, Speed, and Distance

78. Speed = DistanceTime\frac{\text{Distance}}{\text{Time}}
79. Distance = Speed×Time\text{Speed} \times \text{Time}
80. Time = DistanceSpeed\frac{\text{Distance}}{\text{Speed}}
81. Relative speed = Sum or difference of individual speeds\text{Sum or difference of individual speeds}

💸 Commercial Mathematics

82. Simple Interest = P×R×T100\frac{P \times R \times T}{100}
83. Compound Interest = P(1+R100)T−PP(1 + \frac{R}{100})^T – P
84. Profit = Selling Price – Cost Price
85. Loss = Cost Price – Selling Price
86. Discount = Marked Price – Selling Price
87. Percentage = PartWhole×100\frac{\text{Part}}{\text{Whole}} \times 100

📈 Algebraic Identities (More)

88. x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 – xy + y^2)
89. an×am=an+ma^n \times a^m = a^{n+m}
90. anam=an−m\frac{a^n}{a^m} = a^{n-m}
91. (am)n=amn(a^m)^n = a^{mn}
92. (ab)n=anbn(ab)^n = a^n b^n

⛏ Miscellaneous Formulas (Quick Use)

93. Sum of first n natural numbers = n(n+1)2\frac{n(n+1)}{2}
94. Sum of squares of first n numbers = n(n+1)(2n+1)6\frac{n(n+1)(2n+1)}{6}
95. Sum of cubes of first n numbers = (n(n+1)2)2\left(\frac{n(n+1)}{2}\right)^2
96. a×b\sqrt{a \times b} = Geometric mean of a and b
97. 2aba+b\frac{2ab}{a + b} = Harmonic mean of a and b
98. Area of rhombus = 12×d1×d2\frac{1}{2} \times d_1 \times d_2
99. Volume of hemisphere = 23πr3\frac{2}{3}\pi r^3
100. Surface area of hemisphere = 3πr23\pi r^2
101. Area of ellipse = πab\pi ab
102. Volume of prism = base area × height
103. Volume of pyramid = 13×Base Area×h\frac{1}{3} \times \text{Base Area} \times h
104. Surface area of sphere = 4πr24\pi r^2
105. Perimeter of semicircle = πr+2r\pi r + 2r
106. Compound ratio of a:b and c:d = ac : bd
107. Square root of a fraction = ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
108. (a+b+c)2=a2+b2+c2+2(ab+bc+ca)(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
109. (a−b−c)2=a2+b2+c2−2ab+2bc−2ca(a – b – c)^2 = a^2 + b^2 + c^2 – 2ab + 2bc – 2ca
110. (a+b)3=a3+b3+3ab(a+b)(a + b)^3 = a^3 + b^3 + 3ab(a + b)
111. (a−b)3=a3−b3−3ab(a−b)(a – b)^3 = a^3 – b^3 – 3ab(a – b)
112. an−bn=(a−b)(an−1+an−2b+…+bn−1)a^n – b^n = (a – b)(a^{n-1} + a^{n-2}b + \ldots + b^{n-1})
113. an+bna^n + b^n (even n) can’t be factorized over reals
114. log⁡(ab)=log⁡a+log⁡b\log(ab) = \log a + \log b
115. log⁡(a/b)=log⁡a−log⁡b\log(a/b) = \log a – \log b
116. log⁡an=nlog⁡a\log a^n = n \log a
117. log⁡ba=log⁡alog⁡b\log_b a = \frac{\log a}{\log b}
118. ex=1+x+x22!+x33!+…e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots
119. ln⁡e=1\ln e = 1
120. ln⁡1=0\ln 1 = 0

⛽ Mensuration (Advanced)

121. Area of sector = θ360×πr2\frac{\theta}{360} \times \pi r^2
122. Arc length = θ360×2πr\frac{\theta}{360} \times 2\pi r
123. Area of ring = π(R2−r2)\pi(R^2 – r^2)
124. Perimeter of parallelogram = 2(a+b)2(a + b)
125. Height of parallelogram = Areabase\frac{\text{Area}}{\text{base}}
126. Apothem of regular polygon = s2tan⁡(π/n)\frac{s}{2\tan(\pi/n)}
127. Area of regular polygon = 12nsa\frac{1}{2}nsa
128. Surface area of cone = πr(l+r)\pi r (l + r)

🔺 Advanced Trigonometry (129–150)

129. sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B
130. cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B
131. tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}
132. sin⁡(90∘−x)=cos⁡x\sin(90^\circ – x) = \cos x
133. cos⁡(90∘−x)=sin⁡x\cos(90^\circ – x) = \sin x
134. tan⁡(90∘−x)=cot⁡x\tan(90^\circ – x) = \cot x
135. sin⁡(2x)=2sin⁡xcos⁡x\sin(2x) = 2\sin x\cos x
136. cos⁡(2x)=1−2sin⁡2x=2cos⁡2x−1\cos(2x) = 1 – 2\sin^2 x = 2\cos^2 x – 1
137. tan⁡(2x)=2tan⁡x1−tan⁡2x\tan(2x) = \frac{2\tan x}{1 – \tan^2 x}
138. Law of sines: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
139. Law of cosines: c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 – 2ab\cos C
140. cot⁡2x+1=csc⁡2x\cot^2 x + 1 = \csc^2 x
141. tan⁡2x+1=sec⁡2x\tan^2 x + 1 = \sec^2 x
142. Angle sum in triangle = 180∘180^\circ
143. Radian = π180×degrees\frac{\pi}{180} \times \text{degrees}
144. Degrees = 180π×radians\frac{180}{\pi} \times \text{radians}
145. sin⁡x≈x\sin x \approx x, cos⁡x≈1−x22\cos x \approx 1 – \frac{x^2}{2} (for small x)
146. Height = Base×tan⁡(θ)\text{Base} \times \tan(\theta)
147. Area using sine: 12absin⁡C\frac{1}{2}ab\sin C
148. Heron’s formula = s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}, where s=a+b+c2s = \frac{a+b+c}{2}
149. Angle in semicircle = 90∘90^\circ
150. Circle theorem: Angle subtended by same arc is equal

🧾 Sequences & Series (151–175)

151. Arithmetic Series sum = n2(2a+(n−1)d)\frac{n}{2}(2a + (n – 1)d)
152. nth term of AP = a+(n−1)da + (n – 1)d
153. nth term of GP = arn−1ar^{n-1}
154. Sum of n terms in GP = a(rn−1)r−1\frac{a(r^n – 1)}{r – 1}
155. Infinite GP = a1−r\frac{a}{1 – r}, when ∣r∣<1|r| < 1
156. Sum of n odd numbers = n2n^2
157. Sum of n even numbers = n(n+1)n(n + 1)
158. Fibonacci Sequence: Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}
159. Triangular number = n(n+1)2\frac{n(n+1)}{2}
160. Factorial: n!=n×(n−1)!n! = n \times (n-1)!
161. Double factorial: n!!=n(n−2)(n−4)…n!! = n(n-2)(n-4)…
162. Sum of AP in reverse order = Same as original
163. Square number = n2n^2
164. Cube number = n3n^3
165. Pascal’s Triangle gives binomial coefficients
166. Binomial Theorem: (a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k
167. (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}
168. Series of reciprocal natural numbers: Harmonic Series
169. Arithmetic mean ≥ Geometric mean ≥ Harmonic mean
170. Number of subsets = 2n2^n
171. Number of proper subsets = 2n−12^n – 1
172. Sum of all subsets = (1+a1)(1+a2)…(1+an)(1 + a_1)(1 + a_2)…(1 + a_n)
173. Number of ways to choose r items = (nr)\binom{n}{r}
174. Number of permutations = n!n!
175. Circular permutation = (n−1)!(n-1)!

🧠 Logical & Reasoning Based (176–200)

176. If ax=ba^x = b, then x=log⁡abx = \log_a b
177. Truth table: AND = A∧BA \land B, OR = A∨BA \lor B
178. XOR logic = A⊕BA \oplus B = true if A ≠ B
179. De Morgan’s Law: ¬(A∧B)=¬A∨¬B\neg (A \land B) = \neg A \lor \neg B
180. Negation of implication: ¬(A⇒B)=A∧¬B\neg (A \Rightarrow B) = A \land \neg B
181. Contrapositive of A⇒BA \Rightarrow B is ¬B⇒¬A\neg B \Rightarrow \neg A
182. Set union: A∪BA \cup B
183. Set intersection: A∩BA \cap B
184. Set difference: A−BA – B
185. Complement: AcA^c = elements not in A
186. Cartesian Product: A×BA \times B
187. Modulo operation: amod  ba \mod b
188. Floor function: ⌊x⌋\lfloor x \rfloor
189. Ceiling function: ⌈x⌉\lceil x \rceil
190. Absolute value: ∣x∣|x|
191. If ax+by=cax + by = c, it’s a linear Diophantine equation
192. Pythagorean triplets: a2+b2=c2a^2 + b^2 = c^2
193. Euler’s formula: V−E+F=2V – E + F = 2 (polyhedra)
194. Prime factorization = Breaking into product of primes
195. Check for divisibility by 9: Sum of digits ÷ 9
196. Average = Total sumNumber of terms\frac{\text{Total sum}}{\text{Number of terms}}
197. Rate = Quantity ÷ Time
198. Ratio = Comparison of two values
199. Proportion: a:b=c:d⇒ad=bca:b = c:d \Rightarrow ad = bc
200. Percentage change = New – OldOld×100\frac{\text{New – Old}}{\text{Old}} \times 100