Top 100 Most Important Math Formulas

${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$
Description: The square of the sum of two terms.
${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$
Description: The square of the difference of two terms.
$a^{2} - b^{2} = (a + b) (a - b)$
Description: The formula for the difference of two squares.
${(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$
Description: The cube of the sum of two terms.
${(a - b)}^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$
Description: The cube of the difference of two terms.
$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$
Description: The formula for the sum of two cubes.
$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
Description: The formula for the difference of two cubes.
$a^{3} + b^{3} + c^{3} - 3 a b c = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a)$
Description: Relationship between the sum of three cubes and their product.
If a + b + c = 0, then:
$a^{3} + b^{3} + c^{3} = 3 a b c$
Description: A special case of the formula above.
Solution for a quadratic equation (ax² + bx + c = 0):
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
Description: Known as the quadratic formula.
Sum of roots of a quadratic equation:
$α + β = - \frac{b}{a}$
Description: The relationship between the roots and coefficients.
Product of roots of a quadratic equation:
$α β = \frac{c}{a}$
Description: The relationship between the roots and coefficients.
Average:
$Average = \frac{Sum of all items}{Number of items}$
Description: The formula to calculate the average.
Percentage:
$Percentage (%) = (\frac{Value}{Total Value}) \times 100$
Description: The formula to calculate percentage.
Profit:
$Profit = Selling Price - Cost Price$
Description: The formula to calculate profit.
Loss:
$Loss = Cost Price - Selling Price$
Description: The formula to calculate loss.
Profit Percentage:
$Profit % = (\frac{Profit}{Cost Price}) \times 100$
Description: The formula to calculate profit percentage.
Loss Percentage:
$Loss % = (\frac{Loss}{Cost Price}) \times 100$
Description: The formula to calculate loss percentage.
Simple Interest (SI):
$SI = \frac{P \times R \times T}{100}$
Description: Formula for simple interest (P=Principal, R=Rate, T=Time).
Compound Interest (CI):
$CI = P {(1 + \frac{R}{100})}^{T} - P$
Description: The formula to calculate compound interest.
Speed:
$Speed = \frac{Distance}{Time}$
Description: The relationship between speed, distance, and time.
Average Speed (for equal distances):
$Average Speed = \frac{2 x y}{x + y}$
Description: Average speed when traveling equal distances at speeds x and y.
Time taken working together:
$Time together = \frac{x y}{x + y}$
Description: If A can do a job in x days and B in y days, this is the time they take together.
Rule of Proportion:

If a:b::c:d, then
$a d = b c$
Description: The product of extremes equals the product of means.
LCM & HCF Relationship:
$LCM \times HCF = Product of the two numbers$
Description: The relationship between the LCM and HCF of two numbers.
Area of a Square:
$A = {(Side)}^{2}$
Description: The formula to calculate the area of a square.
Perimeter of a Square:
$P = 4 \times Side$
Description: The formula to calculate the perimeter of a square.
Diagonal of a Square:
$d = Side \times \sqrt{2}$
Description: The formula to calculate the diagonal of a square.
Area of a Rectangle:
$A = Length \times Width$
Description: The formula to calculate the area of a rectangle.
Perimeter of a Rectangle:
$P = 2 (Length + Width)$
Description: The formula to calculate the perimeter of a rectangle.
Diagonal of a Rectangle:
$d = \sqrt{{(Length)}^{2} + {(Width)}^{2}}$
Description: The formula to calculate the diagonal of a rectangle.
Area of a Triangle:
$A = \frac{1}{2} \times Base \times Height$
Description: The general formula for the area of a triangle.
Area of an Equilateral Triangle:
$A = \frac{\sqrt{3}}{4} \times {(Side)}^{2}$
Description: The area of a triangle with all three sides equal.
Pythagorean Theorem:
${(Hypotenuse)}^{2} = {(Base)}^{2} + {(Perpendicular)}^{2}$
Description: The relationship between the sides of a right-angled triangle.
Area of a Circle:
$A = π r^{2}$
Description: Formula for the area of a circle (r = radius).
Circumference of a Circle:
$C = 2 π r$
Description: The formula for the perimeter of a circle.
Area of a Rhombus:
$A = \frac{1}{2} \times d_{1} \times d_{2}$
Description: Area of a rhombus (d₁, d₂ = diagonals).
Area of a Trapezium:
$A = \frac{1}{2} \times (a + b) \times h$
Description: Area of a trapezium (a, b = parallel sides, h = height).
Volume of a Cube:
$V = {(Side)}^{3}$
Description: The formula to calculate the volume of a cube.
Total Surface Area of a Cube:
$A = 6 \times {(Side)}^{2}$
Description: The formula for the total surface area of a cube.
Volume of a Cuboid:
$V = l \times b \times h$
Description: Volume of a cuboid (l=length, b=breadth, h=height).
Total Surface Area of a Cuboid:
$A = 2 (l b + b h + h l)$
Description: The formula for the total surface area of a cuboid.
Volume of a Cylinder:
$V = π r^{2} h$
Description: Volume of a cylinder (r = radius, h = height).
Curved Surface Area of a Cylinder:
$A = 2 π r h$
Description: The formula for the curved surface area of a cylinder.
Total Surface Area of a Cylinder:
$A = 2 π r (r + h)$
Description: The formula for the total surface area of a cylinder.
Volume of a Cone:
$V = \frac{1}{3} π r^{2} h$
Description: The formula to calculate the volume of a cone.
Curved Surface Area of a Cone:
$A = π r l$
Description: Curved surface area of a cone (l = slant height).
Volume of a Sphere:
$V = \frac{4}{3} π r^{3}$
Description: The formula to calculate the volume of a sphere.
Surface Area of a Sphere:
$A = 4 π r^{2}$
Description: The formula for the surface area of a sphere.
Volume of a Hemisphere:
$V = \frac{2}{3} π r^{3}$
Description: The formula for the volume of a hemisphere.
Total Surface Area of a Hemisphere:
$A = 3 π r^{2}$
Description: The formula for the total surface area of a hemisphere.
$\sin^{2} θ + \cos^{2} θ = 1$
Description: The fundamental trigonometric identity.
$\sec^{2} θ - \tan^{2} θ = 1$
Description: Another important trigonometric identity.
${cosec}^{2} θ - \cot^{2} θ = 1$
Description: The third fundamental trigonometric identity.
$\sin (A + B) = \sin A \cos B + \cos A \sin B$
Description: Sine of the sum of two angles.
$\cos (A + B) = \cos A \cos B - \sin A \sin B$
Description: Cosine of the sum of two angles.
$\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
Description: Tangent of the sum of two angles.
$\sin 2 A = 2 \sin A \cos A$
Description: Double angle formula for sine.
$\cos 2 A = \cos^{2} A - \sin^{2} A$
Description: Double angle formula for cosine.
For Height and Distance problems:
$\tan θ = \frac{Perpendicular}{Base}$
Description: Used for problems involving angle of elevation and depression.
Sum of the first n natural numbers:
$S_{n} = \frac{n (n + 1)}{2}$
Description: The sum of numbers from 1 to n.
Sum of the first n odd numbers:
$S_{n} = n^{2}$
Description: The sum of the first n consecutive odd numbers.
Sum of the first n even numbers:
$S_{n} = n (n + 1)$
Description: The sum of the first n consecutive even numbers.
Sum of the squares of the first n natural numbers:
$S_{n} = \frac{n (n + 1) (2 n + 1)}{6}$
Description: The sum of squares from 1² to n².
Sum of the cubes of the first n natural numbers:
$S_{n} = {[\frac{n (n + 1)}{2}]}^{2}$
Description: The sum of cubes from 1³ to n³.
nth term of an Arithmetic Progression (AP):
$T_{n} = a + (n - 1) d$
Description: Finding the nth term (a=first term, d=common difference).
Sum of n terms of an Arithmetic Progression (AP):
$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
Description: Sum of the first n terms of an AP.
nth term of a Geometric Progression (GP):
$T_{n} = a r^{n - 1}$
Description: Finding the nth term (a=first term, r=common ratio).
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).
Division Algorithm:
$Dividend = (Divisor \times Quotient) + Remainder$
Description: The fundamental rule of division.
LCM of Fractions:
$LCM = \frac{LCM of Numerators}{HCF of Denominators}$
Description: How to find the LCM of two or more fractions.
HCF of Fractions:
$HCF = \frac{HCF of Numerators}{LCM of Denominators}$
Description: How to find the HCF of two or more fractions.
Sum of interior angles of a polygon:
$Sum = (n - 2) \times 180^{°}$
Description: Sum of interior angles of a polygon with n sides.
Each interior angle of a regular polygon:
$Angle = \frac{(n - 2) \times 180^{°}}{n}$
Description: The measure of each interior angle of a regular n-sided polygon.
Number of diagonals in a polygon:
$Number of Diagonals = \frac{n (n - 3)}{2}$
Description: The formula to find the number of diagonals in an n-sided polygon.
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).
Downstream Speed (Boats and Streams):
$S_{downstream} = S_{boat} + S_{stream}$
Description: The speed of a boat traveling with the current.
Upstream Speed (Boats and Streams):
$S_{upstream} = S_{boat} - S_{stream}$
Description: The speed of a boat traveling against the current.
Relative Speed (Same Direction):
$S_{relative} = S_{1} - S_{2}$
Description: Relative speed of two objects moving in the same direction (S₁ > S₂).
Relative Speed (Opposite Direction):
$S_{relative} = S_{1} + S_{2}$
Description: Relative speed of two objects moving in opposite directions.
Probability of an Event:
$P (E) = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes}$
Description: The formula to calculate the probability of an event.
Permutation:
$P_{r}^{none/>} = \frac{n!}{(n - r)!}$
Description: The number of ways to arrange r items from a set of n items.
Combination:
$C_{r}^{none/>} = \frac{n!}{r! (n - r)!}$
Description: The number of ways to select r items from a set of n items.
Angle between hands of a clock:
$θ = | \frac{60 H - 11 M}{2} |$
Description: Angle in degrees between hour (H) and minute (M) hands.
Rule of Indices:
$a^{m} \times a^{n} = a^{m + n}$
Description: Multiplication of exponents with the same base.
Rule of Indices:
$\frac{a^{m}}{a^{n}} = a^{m - n}$
Description: Division of exponents with the same base.
Rule of Logarithms:
$\log_{a} (m n) = \log_{a} m + \log_{a} n$
Description: Logarithm of a product.
Rule of Logarithms:
$\log_{a} (\frac{m}{n}) = \log_{a} m - \log_{a} n$
Description: Logarithm of a quotient.
Rule of Logarithms:
$\log_{a} (m^{n}) = n \log_{a} m$
Description: Logarithm of a power.
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 $(s = \frac{a + b + c}{2})$ .
${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$
Description: The square of the sum of three terms.
Weighted Average:
$Weighted Average = \frac{w_{1} x_{1} + w_{2} x_{2} + ... + w_{n} x_{n}}{w_{1} + w_{2} + ... + w_{n}}$
Description: Where x is the value and w is its corresponding weight.
Profit Sharing Ratio in Partnership:
$\frac{{Profit}_{1}}{{Profit}_{2}} = \frac{{Investment}_{1} \times {Time}_{1}}{{Investment}_{2} \times {Time}_{2}}$
Description: Partners' profits are proportional to the product of their investment and time.
Difference between CI and SI for 2 years:
$Difference = P {(\frac{R}{100})}^{2}$
Description: The difference between Compound and Simple Interest for 2 years for the same Principal (P) and Rate (R).
Difference between CI and SI for 3 years:
$Difference = P {(\frac{R}{100})}^{2} (3 + \frac{R}{100})$
Description: The difference between Compound and Simple Interest for 3 years for the same Principal (P) and Rate (R).
Time for a train to cross a pole or a person:
$Time = \frac{L_{train}}{S_{train}}$
Description: Time taken when a train crosses a stationary object of negligible length (L=length of train, S=speed of train).
Time for a train to cross a platform or a bridge:
$Time = \frac{L_{train} + L_{platform}}{S_{train}}$
Description: Time taken when a train crosses an object of considerable length.
Area of a Sector of a Circle:
$Area = (\frac{θ}{360}) \times π r^{2}$
Description: The area of a sector of a circle, where θ is the central angle in degrees.
Length of an Arc of a Circle:
$Length = (\frac{θ}{360}) \times 2 π r$
Description: The length of an arc of a circle.
Value from a Pie Chart:
$Value = (\frac{Component Angle}{360}) \times Total Value$
Description: The value of a component in a pie chart is proportional to its central angle.

Top 100 Most Important Math Formulas

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Top 100 Most Important Math Formulas

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