Complete Mathematical Formulas Reference Guide

A comprehensive collection of essential mathematical formulas for students from Class 8 through college level

Basic Arithmetic Formulas

Fundamental Operations

Operation	Formula	Example
Addition	a + b	5 + 3 = 8
Subtraction	a – b	8 – 3 = 5
Multiplication	a × b or a · b	4 × 3 = 12
Division	a ÷ b or a/b	12 ÷ 3 = 4

Percentage Formulas

Type	Formula	Usage
Percentage	(Part/Whole) × 100	Finding percentage of a number
Percentage Increase	((New Value – Original)/Original) × 100	Calculating growth rate
Percentage Decrease	((Original – New Value)/Original) × 100	Calculating reduction rate

Average and Ratio

Concept	Formula	Description
Average (Mean)	(Sum of all values)/(Number of values)	Central tendency measure
Ratio	a:b = a/b	Comparison between quantities
Proportion	a/b = c/d → ad = bc	Equal ratios

Algebraic Formulas

Basic Algebraic Identities

Identity	Formula	Expansion
Square of Sum	(a + b)²	a² + 2ab + b²
Square of Difference	(a – b)²	a² – 2ab + b²
Difference of Squares	a² – b²	(a + b)(a – b)
Cube of Sum	(a + b)³	a³ + 3a²b + 3ab² + b³
Cube of Difference	(a – b)³	a³ – 3a²b + 3ab² – b³
Sum of Cubes	a³ + b³	(a + b)(a² – ab + b²)
Difference of Cubes	a³ – b³	(a – b)(a² + ab + b²)

Quadratic Formulas

Type	Formula	Usage
Standard Form	ax² + bx + c = 0	General quadratic equation
Quadratic Formula	x = (-b ± √(b² – 4ac))/(2a)	Finding roots
Discriminant	Δ = b² – 4ac	Nature of roots
Sum of Roots	α + β = -b/a	Relationship between roots
Product of Roots	αβ = c/a	Relationship between roots

Logarithmic Formulas

Property	Formula	Description
Basic Definition	log_a(x) = y ⟺ a^y = x	Logarithm definition
Product Rule	log_a(xy) = log_a(x) + log_a(y)	Logarithm of product
Quotient Rule	log_a(x/y) = log_a(x) – log_a(y)	Logarithm of quotient
Power Rule	log_a(x^n) = n·log_a(x)	Logarithm of power
Change of Base	log_a(x) = log_b(x)/log_b(a)	Converting bases

Geometry Formulas

2D Shapes – Area and Perimeter

Shape	Area Formula	Perimeter Formula
Rectangle	A = length × width	P = 2(l + w)
Square	A = side²	P = 4 × side
Triangle	A = ½ × base × height	P = a + b + c
Circle	A = πr²	C = 2πr or πd
Parallelogram	A = base × height	P = 2(a + b)
Trapezoid	A = ½(b₁ + b₂) × h	P = a + b₁ + c + b₂
Rhombus	A = ½ × d₁ × d₂	P = 4a

3D Shapes – Surface Area and Volume

Shape	Surface Area	Volume
Cube	SA = 6a²	V = a³
Rectangular Prism	SA = 2(lw + lh + wh)	V = l × w × h
Cylinder	SA = 2πr(r + h)	V = πr²h
Cone	SA = πr(r + l)	V = ⅓πr²h
Sphere	SA = 4πr²	V = ⅔πr³
Pyramid	SA = base area + lateral area	V = ⅓ × base area × height

Special Triangle Formulas

Type	Formula	Application
Pythagorean Theorem	a² + b² = c²	Right triangles
Heron’s Formula	A = √(s(s-a)(s-b)(s-c))	Area using three sides
Semi-perimeter	s = (a + b + c)/2	Used in Heron’s formula

Coordinate Geometry

Distance and Slope Formulas

Concept	Formula	Description
Distance Formula	d = √[(x₂-x₁)² + (y₂-y₁)²]	Distance between two points
Midpoint Formula	M = ((x₁+x₂)/2, (y₁+y₂)/2)	Midpoint of line segment
Slope Formula	m = (y₂-y₁)/(x₂-x₁)	Slope of line
Point-Slope Form	y – y₁ = m(x – x₁)	Equation of line
Slope-Intercept Form	y = mx + b	Linear equation
Standard Form	Ax + By = C	General linear equation

Circle Equations

Type	Formula	Description
Standard Form	(x-h)² + (y-k)² = r²	Circle with center (h,k)
General Form	x² + y² + Dx + Ey + F = 0	Expanded circle equation

Trigonometry Formulas

Basic Trigonometric Ratios

Function	Ratio	Abbreviation
Sine	opposite/hypotenuse	sin θ
Cosine	adjacent/hypotenuse	cos θ
Tangent	opposite/adjacent	tan θ
Cosecant	hypotenuse/opposite	csc θ = 1/sin θ
Secant	hypotenuse/adjacent	sec θ = 1/cos θ
Cotangent	adjacent/opposite	cot θ = 1/tan θ

Trigonometric Identities

Type	Identity
Pythagorean	sin²θ + cos²θ = 1
Pythagorean	1 + tan²θ = sec²θ
Pythagorean	1 + cot²θ = csc²θ
Double Angle	sin(2θ) = 2sin θ cos θ
Double Angle	cos(2θ) = cos²θ – sin²θ
Sum Formula	sin(A + B) = sin A cos B + cos A sin B
Sum Formula	cos(A + B) = cos A cos B – sin A sin B

Standard Angle Values

Angle	0°	30°	45°	60°	90°
sin θ	0	1/2	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	1/2	0
tan θ	0	1/√3	1	√3	∞

Statistics and Probability

Central Tendency

Measure	Formula	Description
Mean	x̄ = (Σx)/n	Average value
Median	Middle value when ordered	Central position
Mode	Most frequently occurring value	Most common value

Dispersion Measures

Measure	Formula	Description
Range	R = Highest – Lowest	Spread of data
Variance	σ² = Σ(x – x̄)²/n	Average squared deviation
Standard Deviation	σ = √(variance)	Square root of variance

Probability Formulas

Type	Formula	Description
Basic Probability	P(E) = (Number of favorable outcomes)/(Total outcomes)	Fundamental probability
Addition Rule	P(A ∪ B) = P(A) + P(B) – P(A ∩ B)	Union of events
Multiplication Rule	P(A ∩ B) = P(A) × P(B	A)
Conditional Probability	P(A	B) = P(A ∩ B)/P(B)

Calculus Formulas

Derivatives

Function	Derivative
Constant	d/dx(c) = 0
Power Rule	d/dx(xⁿ) = nx^(n-1)
Sum Rule	d/dx(f + g) = f’ + g’
Product Rule	d/dx(fg) = f’g + fg’
Quotient Rule	d/dx(f/g) = (f’g – fg’)/g²
Chain Rule	d/dx(f(g(x))) = f'(g(x)) × g'(x)

Common Derivatives

Function	Derivative
sin x	cos x
cos x	-sin x
tan x	sec² x
eˣ	eˣ
ln x	1/x

Integration

Function	Integral
Constant	∫c dx = cx + C
Power Rule	∫xⁿ dx = x^(n+1)/(n+1) + C
Exponential	∫eˣ dx = eˣ + C
Trigonometric	∫sin x dx = -cos x + C
Trigonometric	∫cos x dx = sin x + C

Physics-Related Math Formulas

Motion Formulas

Type	Formula	Variables
Velocity	v = d/t	v = velocity, d = distance, t = time
Acceleration	a = (v₂ – v₁)/t	a = acceleration, v = velocity
Distance	s = ut + ½at²	s = distance, u = initial velocity
Velocity²	v² = u² + 2as	Final velocity equation

Energy and Work

Type	Formula	Description
Kinetic Energy	KE = ½mv²	Energy of motion
Potential Energy	PE = mgh	Energy of position
Work	W = F × d × cos θ	Work done by force
Power	P = W/t	Rate of doing work

Grade-Specific Formula Charts

Class 8 Essential Formulas

Algebra:

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• a² – b² = (a + b)(a – b)

Geometry:

• Area of rectangle = l × w
• Area of square = a²
• Area of triangle = ½ × base × height
• Volume of cube = a³
• Volume of cuboid = l × w × h

Class 9 Key Formulas

Algebra:

• All Class 8 identities plus:
• (x + a)(x + b) = x² + (a + b)x + ab
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Geometry:

• Heron’s formula: A = √(s(s-a)(s-b)(s-c))
• Surface area of sphere = 4πr²
• Volume of sphere = (4/3)πr³

Class 10 Important Formulas

Quadratic Equations:

• ax² + bx + c = 0
• x = (-b ± √(b² – 4ac))/(2a)

Coordinate Geometry:

• Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
• Section formula: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))

Trigonometry:

• sin²θ + cos²θ = 1
• All standard angle values

Quick Reference Tips

Memory Aids

1. SOHCAHTOA for trigonometric ratios
2. PEMDAS for order of operations
3. FOIL for binomial multiplication

Common Mistakes to Avoid

1. Forgetting to apply the quadratic formula correctly
2. Mixing up area and perimeter formulas
3. Incorrect sign handling in algebraic identities
4. Confusing degrees and radians in trigonometry

Frequently Asked Questions (FAQs)

Q. What are the most important basic math formulas every student should know?

The most important basic math formulas include:

• Arithmetic: Percentage formula = (Part/Whole) × 100
• Algebra: (a + b)² = a² + 2ab + b², and a² – b² = (a + b)(a – b)
• Geometry: Area of circle = πr², Area of triangle = ½ × base × height
• Pythagorean Theorem: a² + b² = c²
• Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

These foundational formulas form the basis for advanced mathematical concepts and are essential for academic success from Class 8 through college level.

Q. How can I memorize math formulas quickly and effectively?

To memorize math formulas efficiently:

• Practice regularly: Solve 5-10 problems daily using each formula
• Use flashcards: Write formulas on one side, applications on the other
• Visual learning: Create colorful formula charts and place them where you study
• Understand the logic: Don’t just memorize—understand why the formula works
• Teach others: Explaining formulas to classmates reinforces your memory
• Use mnemonics: Remember SOHCAHTOA for trigonometry (Sine = Opposite/Hypotenuse)
• Apply in real life: Connect formulas to everyday situations for better retention

Research shows that understanding the derivation of a formula helps in long-term retention better than rote memorization.

Q. What is the quadratic formula and when do we use it?

The quadratic formula is: x = (-b ± √(b² – 4ac))/(2a)

This formula is used to find the roots (solutions) of any quadratic equation in the form ax² + bx + c = 0, where:

• a = coefficient of x² (cannot be zero)
• b = coefficient of x
• c = constant term

When to use it:

• When factoring is difficult or impossible
• To find exact solutions for any quadratic equation
• In real-world problems involving projectile motion, area calculations, or profit optimization

The discriminant (b² – 4ac) tells you about the nature of roots:

• Positive: Two real and distinct roots
• Zero: Two real and equal roots
• Negative: No real roots (complex roots)

Q. What are all the algebraic formulas for Class 9 and Class 10?

Essential algebraic formulas for Class 9-10:

Basic Identities:

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• a² – b² = (a + b)(a – b)
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a – b)³ = a³ – 3a²b + 3ab² – b³
• a³ + b³ = (a + b)(a² – ab + b²)
• a³ – b³ = (a – b)(a² + ab + b²)

Additional formulas:

• (x + a)(x + b) = x² + (a + b)x + ab
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Quadratic equations:

• Standard form: ax² + bx + c = 0
• Quadratic formula: x = (-b ± √(b² – 4ac))/(2a)
• Sum of roots: α + β = -b/a
• Product of roots: αβ = c/a

These formulas are crucial for board exams and competitive entrance tests.

Q. What is the distance formula in coordinate geometry and how is it derived?

The distance formula is: d = √[(x₂-x₁)² + (y₂-y₁)²]

This formula calculates the straight-line distance between two points (x₁, y₁) and (x₂, y₂) on a coordinate plane.

Derivation: The formula is derived from the Pythagorean theorem:

1. Plot two points A(x₁, y₁) and B(x₂, y₂) on a coordinate plane
2. Create a right triangle by drawing horizontal and vertical lines
3. Horizontal distance = |x₂ – x₁|
4. Vertical distance = |y₂ – y₁|
5. Apply Pythagorean theorem: d² = (x₂-x₁)² + (y₂-y₁)²
6. Take square root: d = √[(x₂-x₁)² + (y₂-y₁)²]

Practical applications:

• Finding the shortest distance between two cities on a map
• Calculating the length of a line segment
• Solving problems in navigation and GPS technology
• Architecture and engineering measurements

Q. How do I choose which formula to use for a particular math problem?

Step-by-step approach to selecting the right formula:

1. Identify the topic: Determine if it’s algebra, geometry, trigonometry, etc.
2. Read carefully: Understand what the problem is asking for (area, volume, roots, distance, etc.)
3. List known values: Write down all given information
4. Determine unknowns: Identify what you need to find
5. Match with formulas: Choose the formula that connects your known values to the unknown

Topic-specific tips:

• Geometry: Look for keywords like “area,” “perimeter,” “volume,” or “surface area”
• Algebra: Identify if it’s a linear equation, quadratic equation, or polynomial
• Trigonometry: Check if angle measurements or ratios are involved
• Coordinate Geometry: Look for coordinates or points on a plane
• Statistics: Watch for terms like “average,” “mean,” “probability,” or “standard deviation”

Common problem indicators:

• “How far apart?” → Distance formula
• “How much space?” → Area or volume formula
• “At what rate?” → Speed, velocity, or rate formulas
• “What are the solutions?” → Quadratic formula or factoring
• “What is the chance?” → Probability formula

With practice, pattern recognition becomes natural, and you’ll quickly identify which formula applies to each problem type.

Conclusion

This comprehensive formula guide serves as a complete reference for students from Class 8 through college level. Each formula has been carefully verified and presented in a student-friendly format. Regular practice with these formulas will build mathematical confidence and problem-solving skills.