Top 100 Most Important Math Formulas

1. $${(a+b)}^2=a^2+2ab+b^2$$

   Description: The square of the sum of two terms.

2. $${(a-b)}^2=a^2-2ab+b^2$$

   Description: The square of the difference of two terms.

3. $$a^2-b^2=(a+b)(a-b)$$

   Description: The formula for the difference of two squares.

4. $${(a+b)}^3=a^3+3a^{2}b+3a{b}^2+b^3$$

   Description: The cube of the sum of two terms.

5. $${(a-b)}^3=a^3-3a^{2}b+3a{b}^2-b^3$$

   Description: The cube of the difference of two terms.

6. $$a^3+b^3=(a+b)(a^2-ab+b^2)$$

   Description: The formula for the sum of two cubes.

7. $$a^3-b^3=(a-b)(a^2+ab+b^2)$$

   Description: The formula for the difference of two cubes.

8. $$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$

   Description: Relationship between the sum of three cubes and their product.

9. If a + b + c = 0, then:

   $$a^3+b^3+c^3=3abc$$

   Description: A special case of the formula above.

10. Solution for a quadratic equation (ax² + bx + c = 0):

    $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

    Description: Known as the quadratic formula.

11. Sum of roots of a quadratic equation:

    $$\alpha +\beta =-\frac{b}{a}$$

    Description: The relationship between the roots and coefficients.

12. Product of roots of a quadratic equation:

    $$\alpha \beta =\frac{c}{a}$$

    Description: The relationship between the roots and coefficients.

13. Average:

    $$\text{Average}=\frac{\text{Sum of all items}}{\text{Number of items}}$$

    Description: The formula to calculate the average.

14. Percentage:

    $$\text{Percentage (\%)}=\left(\frac{\text{Value}}{\text{Total Value}}\right)\times 100$$

    Description: The formula to calculate percentage.

15. Profit:

    $$\text{Profit}=\text{Selling Price}-\text{Cost Price}$$

    Description: The formula to calculate profit.

16. Loss:

    $$\text{Loss}=\text{Cost Price}-\text{Selling Price}$$

    Description: The formula to calculate loss.

17. Profit Percentage:

    $$\text{Profit \%}=\left(\frac{\text{Profit}}{\text{Cost Price}}\right)\times 100$$

    Description: The formula to calculate profit percentage.

18. Loss Percentage:

    $$\text{Loss \%}=\left(\frac{\text{Loss}}{\text{Cost Price}}\right)\times 100$$

    Description: The formula to calculate loss percentage.

19. Simple Interest (SI):

    $$\mathrm{SI}=\frac{P\times R\times T}{100}$$

    Description: Formula for simple interest (P=Principal, R=Rate, T=Time).

20. Compound Interest (CI):

    $$\mathrm{CI}=P{(1+\frac{R}{100})}^T-P$$

    Description: The formula to calculate compound interest.

21. Speed:

    $$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$$

    Description: The relationship between speed, distance, and time.

22. Average Speed (for equal distances):

    $$\text{Average Speed}=\frac{2xy}{x+y}$$

    Description: Average speed when traveling equal distances at speeds x and y.

23. Time taken working together:

    $$\text{Time together}=\frac{xy}{x+y}$$

    Description: If A can do a job in x days and B in y days, this is the time they take together.

24. Rule of Proportion:

    If a:b::c:d, then

    $$ad=bc$$

    Description: The product of extremes equals the product of means.

25. LCM & HCF Relationship:

    $$\text{LCM}\times \text{HCF}=\text{Product of the two numbers}$$

    Description: The relationship between the LCM and HCF of two numbers.

26. Area of a Square:

    $$A={\left(\text{Side}\right)}^2$$

    Description: The formula to calculate the area of a square.

27. Perimeter of a Square:

    $$P=4\times \text{Side}$$

    Description: The formula to calculate the perimeter of a square.

28. Diagonal of a Square:

    $$d=\text{Side}\times \sqrt{2}$$

    Description: The formula to calculate the diagonal of a square.

29. Area of a Rectangle:

    $$A=\text{Length}\times \text{Width}$$

    Description: The formula to calculate the area of a rectangle.

30. Perimeter of a Rectangle:

    $$P=2(\text{Length}+\text{Width})$$

    Description: The formula to calculate the perimeter of a rectangle.

31. Diagonal of a Rectangle:

    $$d=\sqrt{{\left(\text{Length}\right)}^2+{\left(\text{Width}\right)}^2}$$

    Description: The formula to calculate the diagonal of a rectangle.

32. Area of a Triangle:

    $$A=\frac{1}{2}\times \text{Base}\times \text{Height}$$

    Description: The general formula for the area of a triangle.

33. Area of an Equilateral Triangle:

    $$A=\frac{\sqrt{3}}{4}\times {\left(\text{Side}\right)}^2$$

    Description: The area of a triangle with all three sides equal.

34. Pythagorean Theorem:

    $${\left(\text{Hypotenuse}\right)}^2={\left(\text{Base}\right)}^2+{\left(\text{Perpendicular}\right)}^2$$

    Description: The relationship between the sides of a right-angled triangle.

35. Area of a Circle:

    $$A=\pi r^2$$

    Description: Formula for the area of a circle (r = radius).

36. Circumference of a Circle:

    $$C=2\pi r$$

    Description: The formula for the perimeter of a circle.

37. Area of a Rhombus:

    $$A=\frac{1}{2}\times d_1\times d_2$$

    Description: Area of a rhombus (d₁, d₂ = diagonals).

38. Area of a Trapezium:

    $$A=\frac{1}{2}\times (a+b)\times h$$

    Description: Area of a trapezium (a, b = parallel sides, h = height).

39. Volume of a Cube:

    $$V={\left(\text{Side}\right)}^3$$

    Description: The formula to calculate the volume of a cube.

40. Total Surface Area of a Cube:

    $$A=6\times {\left(\text{Side}\right)}^2$$

    Description: The formula for the total surface area of a cube.

41. Volume of a Cuboid:

    $$V=l\times b\times h$$

    Description: Volume of a cuboid (l=length, b=breadth, h=height).

42. Total Surface Area of a Cuboid:

    $$A=2(lb+bh+hl)$$

    Description: The formula for the total surface area of a cuboid.

43. Volume of a Cylinder:

    $$V=\pi r^{2}h$$

    Description: Volume of a cylinder (r = radius, h = height).

44. Curved Surface Area of a Cylinder:

    $$A=2\pi rh$$

    Description: The formula for the curved surface area of a cylinder.

45. Total Surface Area of a Cylinder:

    $$A=2\pi r(r+h)$$

    Description: The formula for the total surface area of a cylinder.

46. Volume of a Cone:

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

    Description: The formula to calculate the volume of a cone.

47. Curved Surface Area of a Cone:

    $$A=\pi rl$$

    Description: Curved surface area of a cone (l = slant height).

48. Volume of a Sphere:

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

    Description: The formula to calculate the volume of a sphere.

49. Surface Area of a Sphere:

    $$A=4\pi r^2$$

    Description: The formula for the surface area of a sphere.

50. Volume of a Hemisphere:

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

    Description: The formula for the volume of a hemisphere.

51. Total Surface Area of a Hemisphere:

    $$A=3\pi r^2$$

    Description: The formula for the total surface area of a hemisphere.

52. $${\sin }^2\theta +{\cos }^2\theta =1$$

    Description: The fundamental trigonometric identity.

53. $${\sec }^2\theta -{\tan }^2\theta =1$$

    Description: Another important trigonometric identity.

54. $${\mathrm{cosec}}^2\theta -{\cot }^2\theta =1$$

    Description: The third fundamental trigonometric identity.

55. $$\sin (A+B)=\sin A\cos B+\cos A\sin B$$

    Description: Sine of the sum of two angles.

56. $$\cos (A+B)=\cos A\cos B-\sin A\sin B$$

    Description: Cosine of the sum of two angles.

57. $$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$$

    Description: Tangent of the sum of two angles.

58. $$\sin 2A=2\sin A\cos A$$

    Description: Double angle formula for sine.

59. $$\cos 2A={\cos }^{2}A-{\sin }^{2}A$$

    Description: Double angle formula for cosine.

60. For Height and Distance problems:

    $$\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}$$

    Description: Used for problems involving angle of elevation and depression.

61. Sum of the first n natural numbers:

    $$S_n=\frac{n(n+1)}{2}$$

    Description: The sum of numbers from 1 to n.

62. Sum of the first n odd numbers:

    $$S_n=n^2$$

    Description: The sum of the first n consecutive odd numbers.

63. Sum of the first n even numbers:

    $$S_n=n(n+1)$$

    Description: The sum of the first n consecutive even numbers.

64. Sum of the squares of the first n natural numbers:

    $$S_n=\frac{n(n+1)(2n+1)}{6}$$

    Description: The sum of squares from 1² to n².

65. Sum of the cubes of the first n natural numbers:

    $$S_n={\left[\frac{n(n+1)}{2}\right]}^2$$

    Description: The sum of cubes from 1³ to n³.

66. nth term of an Arithmetic Progression (AP):

    $$T_n=a+(n-1)d$$

    Description: Finding the nth term (a=first term, d=common difference).

67. Sum of n terms of an Arithmetic Progression (AP):

    $$S_n=\frac{n}{2}[2a+(n-1)d]$$

    Description: Sum of the first n terms of an AP.

68. nth term of a Geometric Progression (GP):

    $$T_n=a{r}^{n-1}$$

    Description: Finding the nth term (a=first term, r=common ratio).

69. Sum of n terms of a Geometric Progression (GP):

    $$S_n=\frac{a(r^n-1)}{r-1}$$

    Description: Sum of the first n terms of a GP (when r > 1).

70. Division Algorithm:

    $$\text{Dividend}=(\text{Divisor}\times \text{Quotient})+\text{Remainder}$$

    Description: The fundamental rule of division.

71. LCM of Fractions:

    $$\text{LCM}=\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$

    Description: How to find the LCM of two or more fractions.

72. HCF of Fractions:

    $$\text{HCF}=\frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$$

    Description: How to find the HCF of two or more fractions.

73. Sum of interior angles of a polygon:

    $$\text{Sum}=(n-2)\times {180}^{°}$$

    Description: Sum of interior angles of a polygon with n sides.

74. Each interior angle of a regular polygon:

    $$\text{Angle}=\frac{(n-2)\times {180}^{°}}{n}$$

    Description: The measure of each interior angle of a regular n-sided polygon.

75. Number of diagonals in a polygon:

    $$\text{Number of Diagonals}=\frac{n(n-3)}{2}$$

    Description: The formula to find the number of diagonals in an n-sided polygon.

76. Distance between two points in coordinate geometry:

    $$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$$

    Description: The distance formula for points (x₁, y₁) and (x₂, y₂).

77. Downstream Speed (Boats and Streams):

    $$S_{\text{downstream}}=S_{\text{boat}}+S_{\text{stream}}$$

    Description: The speed of a boat traveling with the current.

78. Upstream Speed (Boats and Streams):

    $$S_{\text{upstream}}=S_{\text{boat}}-S_{\text{stream}}$$

    Description: The speed of a boat traveling against the current.

79. Relative Speed (Same Direction):

    $$S_{\text{relative}}=S_1-S_2$$

    Description: Relative speed of two objects moving in the same direction (S₁ > S₂).

80. Relative Speed (Opposite Direction):

    $$S_{\text{relative}}=S_1+S_2$$

    Description: Relative speed of two objects moving in opposite directions.

81. Probability of an Event:

    $$P\left(E\right)=\frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

    Description: The formula to calculate the probability of an event.

82. Permutation:

    $$P_r=\frac{n!}{(n-r)!}$$

    Description: The number of ways to arrange r items from a set of n items.

83. Combination:

    $$C_r=\frac{n!}{r!(n-r)!}$$

    Description: The number of ways to select r items from a set of n items.

84. Angle between hands of a clock:

    $$\theta =\left|\frac{60H-11M}{2}\right|$$

    Description: Angle in degrees between hour (H) and minute (M) hands.

85. Rule of Indices:

    $$a^m\times a^n=a^{m+n}$$

    Description: Multiplication of exponents with the same base.

86. Rule of Indices:

    $$\frac{a^m}{a^n}=a^{m-n}$$

    Description: Division of exponents with the same base.

87. Rule of Logarithms:

    $${\log }_a\left(mn\right)={\log }_{a}m+{\log }_{a}n$$

    Description: Logarithm of a product.

88. Rule of Logarithms:

    $${\log }_a\left(\frac{m}{n}\right)={\log }_{a}m-{\log }_{a}n$$

    Description: Logarithm of a quotient.

89. Rule of Logarithms:

    $${\log }_a\left(m^n\right)=n{\log }_{a}m$$

    Description: Logarithm of a power.

90. Area of a Scalene Triangle (Heron's Formula):

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

    Description: Where a, b, c are the sides and s is the semi-perimeter $(s=\frac{a+b+c}{2})$.

91. $${(a+b+c)}^2=a^2+b^2+c^2+2(ab+bc+ca)$$

    Description: The square of the sum of three terms.

92. Weighted Average:

    $$\text{Weighted Average}=\frac{w_1{x}_1+w_2{x}_2+...+w_n{x}_n}{w_1+w_2+...+w_n}$$

    Description: Where x is the value and w is its corresponding weight.

93. Profit Sharing Ratio in Partnership:

    $$\frac{{\text{Profit}}_1}{{\text{Profit}}_2}=\frac{{\text{Investment}}_1\times {\text{Time}}_1}{{\text{Investment}}_2\times {\text{Time}}_2}$$

    Description: Partners' profits are proportional to the product of their investment and time.

94. Difference between CI and SI for 2 years:

    $$\text{Difference}=P{\left(\frac{R}{100}\right)}^2$$

    Description: The difference between Compound and Simple Interest for 2 years for the same Principal (P) and Rate (R).

95. Difference between CI and SI for 3 years:

    $$\text{Difference}=P{\left(\frac{R}{100}\right)}^2(3+\frac{R}{100})$$

    Description: The difference between Compound and Simple Interest for 3 years for the same Principal (P) and Rate (R).

96. Time for a train to cross a pole or a person:

    $$\text{Time}=\frac{L_{\text{train}}}{S_{\text{train}}}$$

    Description: Time taken when a train crosses a stationary object of negligible length (L=length of train, S=speed of train).

97. Time for a train to cross a platform or a bridge:

    $$\text{Time}=\frac{L_{\text{train}}+L_{\text{platform}}}{S_{\text{train}}}$$

    Description: Time taken when a train crosses an object of considerable length.

98. Area of a Sector of a Circle:

    $$\text{Area}=\left(\frac{\theta }{360}\right)\times \pi r^2$$

    Description: The area of a sector of a circle, where θ is the central angle in degrees.

99. Length of an Arc of a Circle:

    $$\text{Length}=\left(\frac{\theta }{360}\right)\times 2\pi r$$

    Description: The length of an arc of a circle.

100. Value from a Pie Chart:

     $$\text{Value}=\left(\frac{\text{Component Angle}}{360}\right)\times \text{Total Value}$$

     Description: The value of a component in a pie chart is proportional to its central angle.