100 Essential Math Formulas with Examples and Explanations
Master important mathematics formulas with clear definitions, quick examples, and practical explanations—from arithmetic and algebra to geometry, trigonometry, statistics, and calculus.
100 formulasSolved examplesBeginner-friendlyMathematics becomes easier when you understand what a formula means and when to use it. This reference guide brings together 100 commonly used formulas for school study, revision, homework, and introductory competitive-exam preparation.
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- Choose a topic from the table of contents.
- Read the formula and the meaning of its symbols.
- Follow the worked example, then try a similar question yourself.
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Table of Contents
Showing all 100 formulas. Search filters the formula cards below.
1. Arithmetic and Number Basics
Arithmetic formulas help with everyday calculations, divisibility, averages, and the relationship between numbers.
1. Addition
Sum = a + b Add two or more quantities to find their total.
Example: 18 + 7 = 25.
2. Subtraction
Difference = a − b Subtract the second quantity from the first.
Example: 20 − 6 = 14.
3. Multiplication
Product = a × b Multiplication is repeated addition for whole-number factors.
Example: 8 × 6 = 48.
4. Division
Quotient = a ÷ b, where b ≠ 0 Division shares a quantity into equal groups or finds how many times one quantity fits into another.
Example: 42 ÷ 7 = 6.
5. Arithmetic Mean (Average)
Mean = Sum of observations ÷ Number of observations
Example: The mean of 4, 6, and 8 is (4 + 6 + 8) ÷ 3 = 6.
6. HCF–LCM Relationship
HCF(a, b) × LCM(a, b) = a × b This relationship applies to two positive integers.
Example: For 12 and 18, HCF = 6 and LCM = 36; 6 × 36 = 12 × 18 = 216.
7. Speed
Speed = Distance ÷ Time Use consistent units, such as kilometres and hours.
Example: 150 km in 3 hours gives an average speed of 150 ÷ 3 = 50 km/h.
8. Distance
Distance = Speed × Time
Example: At 60 km/h for 2.5 hours, distance = 60 × 2.5 = 150 km.
2. Percentages, Ratio and Interest
These formulas are useful for discounts, marks, finance, comparisons, and proportion problems.
9. Percentage of a Quantity
Percentage amount = (Percentage ÷ 100) × Total
Example: 20% of 250 = (20 ÷ 100) × 250 = 50.
10. Percentage Change
Percentage change = (Change ÷ Original value) × 100% Use a positive change for an increase and a negative change for a decrease.
Example: A price rises from 80 to 100. Change = 20, so increase = (20 ÷ 80) × 100% = 25%.
11. Ratio
Ratio of a to b = a : b = a/b, when b ≠ 0 Simplify ratios by dividing each term by their common factor.
Example: 12 : 18 simplifies to 2 : 3.
12. Proportion
If a/b = c/d, then ad = bc (b,d ≠ 0)
Example: x/6 = 4/3, so 3x = 24 and x = 8.
13. Simple Interest
SI = (P × R × T) ÷ 100 P is principal, R is annual interest rate in percent, and T is time in years.
Example: P = ₹5,000, R = 4%, T = 2 years: SI = ₹400.
14. Amount with Simple Interest
Amount = P + SI
Example: ₹5,000 principal + ₹400 interest = ₹5,400.
15. Compound Amount (Annual Compounding)
A = P(1 + R/100)T For annual compounding, P is principal, R is annual rate in percent, and T is years. Compound interest is A − P.
Example: ₹1,000 at 10% for 2 years gives A = 1000(1.1)² = ₹1,210; interest = ₹210.
16. Profit Percentage
Profit = Selling Price − Cost Price; Profit % = (Profit ÷ Cost Price) × 100%
Example: Cost ₹400, selling price ₹500: profit = ₹100, profit percentage = 25%.
17. Loss Percentage
Loss = Cost Price − Selling Price; Loss % = (Loss ÷ Cost Price) × 100%
Example: Cost ₹500, selling price ₹450: loss = ₹50, loss percentage = 10%.
18. Discount Percentage
Discount % = (Discount ÷ Marked Price) × 100%
Example: Marked price ₹800, sale price ₹600: discount = ₹200, or 25%.
19. Direct Proportion
y = kx When two quantities are directly proportional, their ratio y/x remains constant (k).
Example: If 3 notebooks cost ₹90, then 5 at the same unit price cost ₹150.
3. Algebra and Equations
Algebra uses letters to represent numbers. These identities and equation rules help simplify expressions and solve unknowns.
20. Square of a Sum
(a + b)² = a² + 2ab + b²
Example: (x + 3)² = x² + 6x + 9.
21. Square of a Difference
(a − b)² = a² − 2ab + b²
Example: (x − 4)² = x² − 8x + 16.
22. Difference of Two Squares
a² − b² = (a − b)(a + b)
Example: x² − 25 = (x − 5)(x + 5).
23. Cube of a Sum
(a + b)³ = a³ + 3a²b + 3ab² + b³
Example: (x + 1)³ = x³ + 3x² + 3x + 1.
24. Cube of a Difference
(a − b)³ = a³ − 3a²b + 3ab² − b³
Example: (x − 2)³ = x³ − 6x² + 12x − 8.
25. Sum of Two Cubes
a³ + b³ = (a + b)(a² − ab + b²)
Example: x³ + 8 = (x + 2)(x² − 2x + 4).
26. Difference of Two Cubes
a³ − b³ = (a − b)(a² + ab + b²)
Example: x³ − 27 = (x − 3)(x² + 3x + 9).
27. Linear Equation
ax + b = c ⟹ x = (c − b)/a, where a ≠ 0
Example: 3x + 5 = 20 ⟹ x = (20 − 5)/3 = 5.
28. Quadratic Formula
x = [−b ± √(b² − 4ac)] ÷ (2a), where ax² + bx + c = 0 and a ≠ 0 The expression b² − 4ac is called the discriminant.
Example: x² − 5x + 6 = 0 gives x = [5 ± √1]/2, so x = 2 or 3.
29. Sum and Product of Quadratic Roots
For ax² + bx + c = 0: sum = −b/a; product = c/a
Example: For x² − 5x + 6 = 0, the roots sum to 5 and multiply to 6.
30. Slope-Intercept Form of a Line
y = mx + c m is the slope and c is the y-intercept.
Example: y = 2x + 1 has slope 2 and y-intercept 1.
31. Point-Slope Form
y − y₁ = m(x − x₁)
Example: A line with slope 3 through (2, 5) is y − 5 = 3(x − 2).
32. Two Linear Equations by Cramer's Rule
For ax + by = e and cx + dy = f: x = (ed − bf)/(ad − bc); y = (af − ec)/(ad − bc) These formulas require ad − bc ≠ 0.
Example: x + y = 7 and x − y = 1 give x = 4 and y = 3.
33. Absolute Value
|x| = x if x ≥ 0; |x| = −x if x < 0 Absolute value is the distance from zero on the number line.
Example: |−9| = 9.
4. Exponents, Roots and Logarithms
34. Product of Powers
aᵐ × aⁿ = aᵐ⁺ⁿ
Example: 2³ × 2² = 2⁵ = 32.
35. Quotient of Powers
aᵐ ÷ aⁿ = aᵐ⁻ⁿ, for a ≠ 0
Example: 5⁴ ÷ 5² = 5² = 25.
36. Power of a Power
(aᵐ)ⁿ = aᵐⁿ
Example: (2³)² = 2⁶ = 64.
37. Zero Exponent
a⁰ = 1, for a ≠ 0
Example: 7⁰ = 1.
38. Negative Exponent
a⁻ⁿ = 1/aⁿ, for a ≠ 0
Example: 2⁻³ = 1/8.
39. Square Root
√(a²) = |a| for real a
Example: √(5²) = 5; more generally, √(−5) is not a real number.
40. Logarithm of a Product
logₐ(xy) = logₐx + logₐy Here a > 0, a ≠ 1, and x,y > 0.
Example: log₁₀(100 × 10) = 2 + 1 = 3.
41. Logarithm of a Quotient
logₐ(x/y) = logₐx − logₐy
Example: log₁₀(1000/10) = 3 − 1 = 2.
42. Change of Base for Logarithms
logₐx = logᵦx ÷ logᵦa
Example: log₂8 = log₁₀8 ÷ log₁₀2 = 3.
5. Geometry and 2D Shapes
In the formulas below, π is approximately 3.14159. Use consistent units and remember that area is measured in square units.
43. Area of a Rectangle
A = l × w
Example: Length 8 cm and width 3 cm: A = 24 cm².
44. Perimeter of a Rectangle
P = 2(l + w)
Example: l = 8 cm, w = 3 cm: P = 22 cm.
45. Area of a Square
A = s²
Example: Side 5 cm: A = 25 cm².
46. Perimeter of a Square
P = 4s
Example: Side 5 cm: P = 20 cm.
47. Area of a Triangle
A = ½ × b × h
Example: Base 10 cm, height 6 cm: A = 30 cm².
48. Heron's Formula
A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2 Use a, b, and c as the side lengths of a triangle.
Example: For sides 3, 4, 5, s = 6, so A = √(6 × 3 × 2 × 1) = 6 square units.
49. Area of a Parallelogram
A = b × h
Example: Base 9 cm and perpendicular height 4 cm: A = 36 cm².
50. Area of a Trapezoid (Trapezium)
A = ½(a + b)h Here a and b are the parallel sides and h is the perpendicular height.
Example: Parallel sides 8 cm and 12 cm, height 5 cm: A = 50 cm².
51. Circumference of a Circle
C = 2πr = πd
Example: Radius 7 cm: C = 14π cm ≈ 43.98 cm.
52. Area of a Circle
A = πr²
Example: Radius 7 cm: A = 49π cm² ≈ 153.94 cm².
53. Arc Length (Angle in Degrees)
Arc length = (θ/360°) × 2πr
Example: A 90° arc of a circle with radius 4 cm has length = 2π cm.
54. Area of a Sector (Angle in Degrees)
Sector area = (θ/360°) × πr²
Example: A 90° sector with radius 4 cm has area = 4π cm².
55. Sum of Interior Angles of a Polygon
Sum = (n − 2) × 180° n is the number of sides.
Example: A hexagon has (6 − 2) × 180° = 720° total interior angle sum.
56. Each Interior Angle of a Regular Polygon
Each angle = [(n − 2) × 180°] ÷ n
Example: A regular hexagon has each interior angle = 120°.
57. Pythagorean Theorem
a² + b² = c² For a right triangle, c is the hypotenuse, opposite the right angle.
Example: Legs 3 and 4 give c = √(9 + 16) = 5.
58. Sum of Angles in a Triangle
A + B + C = 180°
Example: If two angles are 50° and 60°, the third is 70°.
59. Distance Between Two Points
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Example: Between (0,0) and (3,4), d = 5 units.
6. 3D Shapes and Mensuration
Surface area is measured in square units; volume is measured in cubic units.
60. Volume of a Cube
V = a³
Example: Side 3 cm: V = 27 cm³.
61. Total Surface Area of a Cube
TSA = 6a²
Example: Side 3 cm: TSA = 54 cm².
62. Volume of a Cuboid
V = lwh
Example: 4 cm × 3 cm × 2 cm = 24 cm³.
63. Total Surface Area of a Cuboid
TSA = 2(lw + lh + wh)
Example: l = 4, w = 3, h = 2 cm: TSA = 2(12 + 8 + 6) = 52 cm².
64. Volume of a Cylinder
V = πr²h
Example: r = 2 cm, h = 5 cm: V = 20π cm³.
65. Curved Surface Area of a Cylinder
CSA = 2πrh
Example: r = 2 cm, h = 5 cm: CSA = 20π cm².
66. Total Surface Area of a Closed Cylinder
TSA = 2πr(h + r)
Example: r = 2 cm, h = 5 cm: TSA = 28π cm².
67. Volume of a Sphere
V = (4/3)πr³
Example: r = 3 cm: V = 36π cm³.
68. Surface Area of a Sphere
A = 4πr²
Example: r = 3 cm: A = 36π cm².
69. Volume of a Cone
V = (1/3)πr²h
Example: r = 3 cm, h = 4 cm: V = 12π cm³.
70. Curved Surface Area of a Cone
CSA = πrl l is the slant height.
Example: r = 3 cm, l = 5 cm: CSA = 15π cm².
71. Total Surface Area of a Cone
TSA = πr(l + r)
Example: r = 3 cm, l = 5 cm: TSA = 24π cm².
72. Volume of a Prism
V = Base area × Perpendicular height
Example: Base area 12 cm² and height 7 cm: V = 84 cm³.
7. Coordinate Geometry
73. Midpoint Formula
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Example: Midpoint of (2,4) and (6,8) is (4,6).
74. Slope of a Line
m = (y₂ − y₁)/(x₂ − x₁), when x₂ ≠ x₁
Example: From (1,2) to (3,8), slope = (8 − 2)/(3 − 1) = 3.
75. Coordinate Distance Formula
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Example: From (1,1) to (4,5), d = √(9 + 16) = 5.
76. Equation of a Circle
(x − h)² + (y − k)² = r² The centre is (h,k) and radius is r.
Example: A circle centred at (2, −1) with radius 3 has equation (x − 2)² + (y + 1)² = 9.
8. Trigonometry
In right-triangle trigonometry, θ is an acute angle. The opposite and adjacent sides are named relative to θ; the hypotenuse is opposite the right angle.
77. Sine Ratio
sin θ = Opposite ÷ Hypotenuse
Example: If opposite = 3 and hypotenuse = 5, sin θ = 3/5.
78. Cosine Ratio
cos θ = Adjacent ÷ Hypotenuse
Example: If adjacent = 4 and hypotenuse = 5, cos θ = 4/5.
79. Tangent Ratio
tan θ = Opposite ÷ Adjacent
Example: If opposite = 3 and adjacent = 4, tan θ = 3/4.
80. Fundamental Trigonometric Identity
sin²θ + cos²θ = 1
Example: If sin θ = 3/5 for an acute angle, cos θ = 4/5 and 9/25 + 16/25 = 1.
81. Tangent Identity
tan θ = sin θ ÷ cos θ, when cos θ ≠ 0
Example: For sin θ = 3/5 and cos θ = 4/5, tan θ = 3/4.
82. Reciprocal Trigonometric Ratios
cosec θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ These expressions are defined only where their denominators are nonzero.
Example: If cos θ = 4/5, then sec θ = 5/4.
83. Sine of a Sum
sin(A + B) = sin A cos B + cos A sin B
Example: sin(30° + 60°) = (1/2)(1/2) + (√3/2)(√3/2) = 1.
84. Cosine of a Sum
cos(A + B) = cos A cos B − sin A sin B
Example: cos(30° + 60°) = 1/4 − 3/4 = −1/2.
85. Sine Double-Angle Formula
sin 2θ = 2 sin θ cos θ
Example: If sin θ = 3/5 and cos θ = 4/5, then sin 2θ = 24/25.
86. Cosine Double-Angle Formula
cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
Example: If sin θ = 3/5 and cos θ = 4/5, cos 2θ = 16/25 − 9/25 = 7/25.
9. Sequences and Series
87. nth Term of an Arithmetic Sequence
aₙ = a₁ + (n − 1)d a₁ is the first term and d is the common difference.
Example: For 3, 7, 11, 15, …, a₅ = 3 + 4 × 4 = 19.
88. Sum of an Arithmetic Series
Sₙ = n(a₁ + aₙ)/2
Example: Sum of 2 + 4 + 6 + 8 + 10 = 5(2 + 10)/2 = 30.
89. nth Term of a Geometric Sequence
aₙ = a₁rⁿ⁻¹ r is the common ratio.
Example: For 2, 6, 18, 54, …, a₄ = 2 × 3³ = 54.
90. Sum of a Finite Geometric Series
Sₙ = a₁(1 − rⁿ)/(1 − r), when r ≠ 1
Example: 2 + 6 + 18 = 2(1 − 3³)/(1 − 3) = 26.
10. Statistics and Probability
91. Mean of a Data Set
Mean = (Σx)/n Add all observations and divide by the number of observations.
Example: For 2, 5, 8, mean = 15/3 = 5.
92. Probability of an Event (Equally Likely Outcomes)
P(E) = Number of favourable outcomes ÷ Total number of outcomes
Example: For a fair six-sided die, P(rolling a 4) = 1/6.
93. Complement Rule
P(not E) = 1 − P(E)
Example: If P(rain) = 0.3, then P(no rain) = 0.7 under the same model.
94. Addition Rule for Mutually Exclusive Events
P(A or B) = P(A) + P(B) This version applies when A and B cannot happen together.
Example: On one fair die, P(1 or 2) = 1/6 + 1/6 = 1/3.
95. Multiplication Rule for Independent Events
P(A and B) = P(A) × P(B)
Example: Two independent coin tosses both show heads with probability 1/2 × 1/2 = 1/4.
11. Introductory Calculus
These introductory formulas assume the relevant functions are defined and differentiable or integrable on the domain being considered.
96. Power Rule for Differentiation
d(xⁿ)/dx = nxⁿ⁻¹
Example: d(x⁴)/dx = 4x³.
97. Derivative of a Constant
d(c)/dx = 0 A constant does not change as x changes.
Example: d(12)/dx = 0.
98. Sum Rule for Differentiation
d[f(x) + g(x)]/dx = f′(x) + g′(x)
Example: d(x² + x³)/dx = 2x + 3x².
99. Power Rule for Integration
∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1 C is the constant of integration.
Example: ∫x² dx = x³/3 + C.
100. Definite Integral of a Power
∫ₐᵇ xⁿ dx = [bⁿ⁺¹ − aⁿ⁺¹]/(n + 1), for n ≠ −1 This evaluates the definite integral for real bounds where the power function is integrable.
Example: ∫₀² x dx = (2² − 0²)/2 = 2.
Tips for Learning Math Formulas
- Understand before memorising: know what each symbol represents and why the formula works.
- Practise substitution: write down the known values, substitute carefully, and keep units consistent.
- Draw diagrams: geometry and trigonometry are easier when you label the shape or triangle.
- Revise in short sessions: review a small group of formulas daily and test yourself without looking.
- Check your result: estimate the answer and confirm that units and signs make sense.
Frequently Asked Questions (FAQs)
1. What are the most important math formulas to learn first?
Start with arithmetic operations, percentages, fractions, ratios, algebraic identities, area and perimeter, the Pythagorean theorem, and basic probability. Choose formulas that match your current class or exam syllabus.
2. How can I memorise math formulas easily?
Understand the meaning of each formula, solve several examples, write the formula from memory, and revise it after increasing intervals. Flashcards and a formula notebook can help.
3. Are these 100 formulas enough for every mathematics exam?
No single list covers every syllabus. This guide is a broad reference for common school topics and introductory calculus. Check your official curriculum and exam requirements for additional formulas.
4. What is the difference between area and perimeter?
Perimeter measures the distance around a two-dimensional shape and uses linear units such as cm. Area measures the surface inside a shape and uses square units such as cm².
5. What does π (pi) mean in mathematics?
Pi is the constant ratio of a circle's circumference to its diameter. It is approximately 3.14159 and appears in circle, sphere, cylinder, and other geometry formulas.
6. Which formulas are useful for competitive exams?
Depending on the exam, useful areas often include percentages, ratio and proportion, profit and loss, simple and compound interest, speed–time–distance, algebra, geometry, trigonometry, and probability. Follow the exam's official syllabus.
7. Can beginners use this math formula guide?
Yes. The arithmetic, percentage, algebra, and basic geometry sections are suitable starting points. Calculus and some trigonometric identities require more background knowledge.
8. How often should I practise formulas?
Practise regularly in short sessions. Try solving questions without checking the formula first, then review any errors and repeat similar problems later.
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