What are the basic formulas of maths?

Question

Infinity Learn · Accepted Answer

Basic math formulas are foundational tools used across arithmetic, algebra, geometry, and trigonometry. These formulas simplify calculations, help in solving mathematical problems efficiently, and form the basis of advanced studies. Here are some of the most important basic formulas you must know:

Arithmetic Formulas:

Sum of n natural numbers = n(n + 1)/2
Sum of squares of first n natural numbers = n(n + 1)(2n + 1)/6

Algebra Formulas (Algebraic Identities):

Formula Type	Formula
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

Geometry Formulas:

Perimeter of a rectangle = 2(l + b)
Area of rectangle = l × b
Area of triangle = (1/2) × base × height
Circumference of a circle = 2πr
Area of circle = πr²

Trigonometry Basics:

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

Arithmetic

Sum of first n natural numbers:
S₁ = 1 + 2 + … + n = n(n + 1) / 2

Sum of squares of first n natural numbers:

S₂ = 1² + 2² + … + n² = n(n + 1)(2n + 1) / 6

Sum of cubes of first n natural numbers:

S₃ = 1³ + 2³ + … + n³ = [n(n + 1) / 2]²

Simple interest:
```
I = P × r × t
```
Compound interest:
```
A = P × (1 + r/n)n·t
```

Algebraic Identities & Sequences

Quadratic formula (roots of ax² + bx + c = 0):
```
x = [–b ± √(b² – 4ac)] / (2a)
```
Difference of squares:
```
a² – b² = (a – b)(a + b)
```

Sum & difference of cubes:

a³ – b³ = (a – b)(a² + ab + b²)
a³ + b³ = (a + b)(a² – ab + b²)

Arithmetic progression:

aₙ = a₁ + (n – 1)d
Sₙ = (n / 2) × [2a₁ + (n – 1)d]

Geometric progression:

aₙ = a₁ × rn – 1
Sₙ = a₁ × (1 – rⁿ) / (1 – r) (r ≠ 1)

Geometry

Triangle area:
A = ½ × base × height
Pythagorean theorem:
```
a² + b² = c²
```
Circle circumference:
```
C = 2πr
```
Circle area:
```
A = πr²
```
Rectangle area:
```
A = length × width
```
Rectangle perimeter:
```
P = 2 × (length + width)
```
Sphere volume:
```
V = (4/3) × πr³
```
Cylinder volume:
```
V = πr² × h
```
Cone volume:
```
V = (1/3) × πr² × h
```

Trigonometry

Fundamental identity:
```
sin²θ + cos²θ = 1
```
Tangent identity:
```
tan θ = sin θ / cos θ
```

Angle sum & difference:

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B

Coordinate Geometry

Distance formula:

d = √[(x₂ – x₁)² + (y₂ – y₁)²]

Midpoint formula:

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Slope of a line:
```
m = (y₂ – y₁) / (x₂ – x₁)
```
Point–slope form:
```
y – y₁ = m (x – x₁)
```

Basic Statistics

Mean (average):
```
μ = (Σ xᵢ) / n
```
Variance:
```
σ² = Σ (xᵢ – μ)² / n
```
Standard deviation:
```
σ = √σ²
```

Knowing these formulas reduces calculation errors, saves time in exams, and strengthens your problem solving abilities. Start by memorizing these in stages and practice with examples for mastery.

What are the basic formulas of maths?

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