**High school math formulas.** Mathematical formulas represent a synthesis of the development of reasoning and are made up of numbers and letters.

Knowing them is necessary to solve many problems that are charged in competitions and in Enem, mainly because it often reduces the time to solve a problem.

However, just decorating the formulas is not enough to be successful in their application. Knowing the meaning of each quantity and understanding the context in which each formula should be used is critical.

In this text we gather the main formulas used in secondary school, grouped by content.

Table of contents

- 1 The functions
- 2 Progressions
- 3 Plane geometry
- 4 Trigonometry
- 5 Combinatorial analysis
- 6 Probability
- 7 Statistics
- 8 Financial mathematics
- 9 Spatial geometry
- 10 Analytic geometry
- 11 Complex numbers

## The functions

Functions represent a relationship between two variables, so a value assigned to one of them will correspond to a unique value of the other.

Two variables can be associated in different ways and according to their formation rule, they receive different classifications.

### Refine function

**f(x) = ax + b**

a: slope

b: linear coefficient

### Quadratic function

**f (x) = ax ^{2}+ bx + c** , where ≠ 0

a, b, and c: second degree function coefficients

#### Roots of the quadratic function

#### Vertex of the parable.

Δ: discriminant of the quadratic equation ( **Δ = b ^{2} - 4.ac**)

a, b and c: coefficients of the quadratic equation

### Modular function

### Exponential function

**f(x) = a ^{x}**, with a> 0 and ≠ 0

### Logarithmic function

**f(x) = log _{un }x** , with positive real and a 1

### Sine function

**f (x) = sin x**

### Cosine function

**f(x) = cos x **

### Polynomial function

**f(x) = a _{n }. x^{n} + a_{n-1. }x^{n-1}+… + A_{2 }. x^{2 }+ a_{1 }. x^{1 }+ a_{0 0}**

un_{n}el_{n-1}, …, he_{2}el_{1}el_{0 0} : complex numbers

n: integer

x: complex variable

## Progressions

Progressions are numerical sequences in which, starting with the first term, all the others are obtained by adding or multiplying by the same value.

In progressions called arithmetic, subsequent terms are found by adding the preceding term with the same number (ratio).

In geometric progressions, the sequence is formed by multiplying the previous term by the ratio.

### Arithmetic progression

#### General term

**un _{n }= A_{1} + (n - 1) r**

un_{n}: General term

un_{1}: 1st term

n: number of terms

r: BP ratio

#### Sum of a finite PA

S_{n}: sum of n terms

un_{1}: 1st term

un_{n}: nth term

n: number of terms

### Geometric Progression

#### General term

**un _{n} = A_{1 }. what^{n-1}**

un_{n}: nth term

un_{1}: 1st term

q: PG ratio

n: number of terms

#### Sum of a finite PG

S_{n}: sum of n terms

un_{1}: 1st term

q: PG ratio

n: number of terms

#### Limit of the sum of an infinite GP

: sum limit when the number of terms tends to infinite

un_{1}: 1st term

q: PG ratio

n: number of terms

See also:

## Plane geometry

Plane geometry is the part of mathematics that studies the properties of geometric figures in the plane. The study of geometry involves the application of postulates, axioms and theorems.

### Sum of the interior angles of a polygon.

**S _{yo} = (n - 2). 180º**

S_{yo}: sum of interior angles

n: number of sides of the polygon

### Story theorem

AB and CD: segments of a line determined by cutting with a bundle of parallel lines

A´B´ and C´D´: segments of another straight line, transversal to the first one, determined by cutting with the same bundle of parallel lines

### Metric relationships in the right triangle

**b ^{2} = a. n**

a: hypotenuse

b: side

n: projection of catheter b over the hypotenuse

**c ^{2} = a. m**

a: hypotenuse

c: side

m: projection of side c on the hypotenuse

**ah = b. c**

a: hypotenuse

b and c: collectors

h: height relative to the hypotenuse

**h ^{2} = m. n**

h: height relative to the hypotenuse

m: projection of side c on the hypotenuse

n: projection of catheter b over the hypotenuse

**un ^{2} = b^{2} + c^{2}** (Pythagoras theorem)

a: hypotenuse

b and c: collectors

### Polygon inscribed in the circumference.

#### Inscribed equilateral triangle

: measured on the side of the inscribed triangle

r: radius of the circumference

r: radius of the circumference

un_{3}: apothem of the inscribed equilateral triangle

#### Registered square

: measured on the side of the inscribed square

r: radius of the circumference

un_{4 4}: apothem of the inscribed square

r: radius of the circumference

#### Inscribed regular hexagon

measure on the side of the inscribed hexagon

r: radius of the circumference

un_{6 6}: insertion of the inscribed hexagon

r: radius of the circumference

### Circumference length

**C = 2.π.r**

C: circumference length

r: radius of the circumference

### Plane figures area

#### Triangle area

A: area of the triangle

b: measure of the base

h: height measurement relative to the base

#### Heron formula for the area of the triangle

p: semiperimeter

a, b and c: sides of the triangle

#### Equilateral triangle area

A: area of the equilateral triangle

measure on the side of the equilateral triangle

#### Rectangle area

**A = bh**

A: rectangular area

b: measure of the base

h: height measurement

#### Square area

**A = L ^{2}**

A: square area

L: side measurement

#### Parallelogram area

**A = bh**

A: area of parallelogram

b: base

h: height

#### Trapezoidal area

A: trapezoidal area

B: measurement of the main base

b: measurement of the smallest base

h: height measurement

#### Rhombus area

A: rhombus area

D: measure of the largest diagonal

d: smallest diagonal measurement

#### Regular area of the hexagon

A: regular hexagon area

lateral hexagon measurement

#### Circle area

**A = π. r ^{2}**

A: area of the circle

r: radius measure

#### Circular sector area

A: area of the circular sector

α_{rad}: angle in radians

A: radius

α_{degrees}: angle in degrees

See more:

## Trigonometry

Trigonometry is the part of mathematics that studies the relationships between the sides and angles of triangles.

It is also used in other areas of study, such as physics, geography, astronomy, engineering, among others.

### Trigonometric relationships

sin: sine of angle B

b: side opposite angle B

a: hypotenuse

cos: cosine of angle B

c: side adjacent to angle B

a: hypotenuse

tg: tangent of angle B

b: side opposite angle B

c: side adjacent to angle B

**sen ^{2} α + cos^{2} α = 1**

sin α: sine of angle α

cos α: cosine of angle α

tg α: tangent of angle α

sin α: sine of angle α

cos α: cosine of angle α

cotg α: cotangent of angle α

tg α: tangent of angle α

sin α: sine of angle α

cos α: cosine of angle α

sec α: secant of angle α

cos α: cosine of angle α

α cossec: angular cosecant α

sin α: sine of angle α

**tg ^{2} α + 1 = sec^{2} α**

tg α: tangent of angle α

sec α: secant of angle α

**cotg ^{2} α + 1 = cosec^{2} α**

cotg α: cotangent of angle α

α cossec: angular cosecant α

### Sine Law

a: side measurement

sin: sine of the angle opposite side a

b: side measurement

sin: sine of angle opposite side b

c: side measurement

sin: sine of the angle opposite side c

### Cosine law

**un ^{2} = b^{2} + c^{2} - 2.bccos **

a, b and c: sides of the triangle

cos: cosine of the angle opposite side a

### Trigonometric transformations

#### Sine of the sum of two arcs

**sin (a + b) = sin a. cos b + sin b.cos a**

sin (a + b): sine of addition of arc a with arc b

without a: sine of arc a

cos b: cosine of arc b

sin b: sine of arc b

cos a: cosine of arc a

#### Sine of the difference of two arches

**sin (a - b) = sin a. cos b - sin b.cos a**

sin (a - b): sine of the subtraction of arc a with arc b

without a: sine of arc a

cos b: cosine of arc b

sin b: sine of arc b

cos a: cosine of arc a

#### Cosine of the sum of two arcs.

**cos (a + b) = cos a. cos b - sin a. sin b**

cos (a + b): cosine of the sum of arc a to arc b

cos a: cosine of arc a

cos b: cosine of arc b

without a: sine of arc a

sin b: sine of arc b

#### Cosine of the difference of two arcs.

**cos (a - b) = cos a. cos b + sin a. sin b**

cos (a - b): cosine of the subtraction of arc a with arc b

cos a: cosine of arc a

cos b: cosine of arc b

without a: sine of arc a

sin b: sine of arc b

#### Tangent of the sum of two arcs.

tg (a + b): tangent of the sum of arc a to arc b (arcs where the tangent is defined)

tg a: tangent of arc a

tg b: tangent of arc b

#### Tangent of the difference of two arcs.

tg (a - b): tangent of the subtraction of arc a with arc b (arcs where the tangent is defined)

tg a: tangent of arc a

tg b: tangent of arc b

#### See more:

## Combinatorial analysis

In combinatorial analysis we study the methods and techniques that allow solving problems related to counting.

The formulas used in this content are often used to solve probability problems.

### Simple permutation

**P = n!**

n !: n. (n - 1) (n - 2)… 3) 2) 1

### Simple fix

### Simple combination

### Newton's binomial

T_{k + 1}: General term

See also Combinatorial analysis exercises.

## Probability

The study of probability allows to obtain the value of possible occurrences in a random experiment (random phenomenon). In other words, probability analyzes the "chances" of obtaining a certain result.

p (A): probability of occurrence of event A

n (A): number of favorable results

n (Ω): number of possible outcomes

### Probability of joining two events.

**p (AUB) = p (A) + p (B) - p (A ∩ B)**

p (AUB): probability of event A or event B occurring

p (A): probability of event A

p (B): probability of event B occurring

p (A ∩ B): probability of event A and event B occurring

### Probability of mutually exclusive events.

**p (AUB) = p (A) + p (B)**

p (AUB): probability of event A or event B occurring

p (A): probability of event A

p (B): probability of event B occurring

### Conditional probability

p (A / B): probability that event A has occurred, event B

p (A ∩ B): probability of event A and event B occurring

p (B): probability of event B occurring

### Probability of independent events.

**p (A ∩ B) = p (A). p (B)**

p (A ∩ B): probability of event A and event B occurring

p (A): probability of event A

p (B): probability of event B occurring

## Statistics

In statistics, we study the collection, recording, organization and analysis of research data.

Using mathematical formulas, it is possible to know the information related to a given population from the data of a sample of that population.

### Arithmetic average

M_{Un}: arithmetic average

: sum of all sample values

n: amount of sample data

### Variance

V: variance

(x_{yo} - M_{Un}): deviation of x values from the arithmetic mean

n: amount of sample data

### Standard deviation

SD: standard deviation

V: variance

See also Statistics and statistics - Exercises

## Financial mathematics

Studying the equivalence of capital over time is the focus of financial mathematics, using formulas that allow us to know how the value of money varies over time.

### Simple interest

**J = C. i. t**

J: interest

C: capital

i: interest rate

t: application time

**M = C + J**

M: quantity

C: capital

J: interest

### Compound swears

**M = C(1 + i) ^{t}**

M. quantity

C: capital

i: interest rate

t: application time

**J = M - C**

J: interest

M: quantity

C: capital

## Spatial geometry

Spatial geometry corresponds to the area of mathematics that is responsible for studying figures in space, that is, those that have more than two dimensions.

### Euler relation

**V - A + F = 2**

V: number of vertices

A: number of edges

F: number of faces

### Prisma

d: diagonal of the paver

a, b and c: measurements of the dimensions of the paver

**V = B.h**

V: prism volume

B: base area

h: height of the prism

### Pyramid

V: volume of the pyramid

B: base area

h: height of the pyramid

#### Pyramidal trunk

V: volume of the pyramidal trunk

h: height of the pyramidal trunk

B: area of the largest base

b: area of the smallest base

### Cylinder

**Un _{L}= 2.π.Rh**

Un_{L}: lateral area

A: radius

h: cylinder height

**Un _{B} = 2.π.R^{2}**

Un_{B}: base area

A: radius

**Un _{T} = 2.π.R (h + R)**

Un_{T}: total area

A: radius

h: height

**V = π.R ^{2}.h**

V: volume

A: radius

### Cone

**Un _{L }= π.R. g**

Un_{L}: lateral area

A: radius

g: generatrix

**Un _{B} = π.R^{2}**

Un_{B}: base area

A: radius

**Un _{T} = π.R. (g + R)**

Un_{T} : total area

A: radius

g: generatrix

V: volume

Un_{B}: base area

h: height

### Cone trunk

**Un _{L} = π.g (R + r)**

Un_{L}: lateral area

g: generatrix

R: major radius

r: smaller radius

V: volume

h: height

R: major radius

r: smaller radius

### Sphere

**A = 4.π.R ^{2}**

A: area of sphere

A: radius

V: volume of sphere

A: radius

See more:

## Analytic geometry

In analytical geometry we represent lines, circles, ellipses, among others in the Cartesian plane. Therefore, it is possible to describe these geometric shapes using equations.

d (A, B): distance between points A and B

x_{1}: abscissa of point A

x_{2}: abscissa of point B

y_{1}: abscissa of point A

y_{2}: abscissa of point B

m: slope of the line

x_{1}: abscissa of point A

x_{2}: abscissa of point B

y_{1}: abscissa of point A

y_{2}: abscissa of point B

### General equation for a straight line.

**ax + by + c = 0**

a, b and c: constants

### Reduced linear equation

**y = m x + b**

m: slope

b: linear coefficient

### Line segmentation equation

a: value at which the line intersects the x-axis

b: value at which the line intersects the y-axis

### Distance between a point and a line

d: distance between point and line

a, b and c: coefficients of the line

x: abscissa point

y: ordinate of the point

### Angle between two lines

m_{1}: slope of line 1

m_{2}: slope of line 2

### Circumference

#### Circumference equation

**(x - x _{c})^{2} + (and - and_{c})^{2} = R^{2}**

x and y: coordinates of any point that belongs to a circle

x_{c }yy_{c}: coordinates of the center of the circle

A: radius

#### Normal equation of circumference

**x ^{2} + Y^{2} - 2.x_{c}.x - 2.y_{c}.y + (x_{c}^{2 }+ Y_{c}^{2} - R^{2}) = 0**

x and y: coordinates of any point that belongs to a circle

x_{c }yy_{c}: coordinates of the center of the circle

A: radius

### Ellipse

(the major axis belongs to the x axis)

x and y: coordinates of any point that belongs to an ellipse

a: measurement of the semi-principal axis

b: measurement of the minor half axis

(the major axis belongs to the y axis)

x and y: coordinates of any point that belongs to an ellipse

a: measurement of the semi-principal axis

b: measurement of the minor half axis

### Hyperbole

(the real axis belongs to the x axis)

x and y: coordinates of any point that belongs to a hyperbola

a: measure of the real semi-axis

b: measure of the imaginary semi-axis

(the real axis belongs to the y axis)

x and y: coordinates of any point that belongs to a hyperbola

a: measure of the real semi-axis

b: measure of the imaginary semi-axis

### Parable

**y ^{2} = 2.px **(vertex at the origin and focus on the abscissa axis)

x and y: coordinates of any point that belongs to the parabola

p: parameter

**x ^{2} = 2.py **(vertex at the origin and focus on the ordinate axis)

x and y: coordinates of any point that belongs to the parabola

p: parameter

## Complex numbers

Complex numbers are numbers made up of a real and imaginary part. The imaginary part is represented by the letter ie indicates the result of the equation i^{2} = -1.

### Algebraic form

**z = a + bi**

z: complex number

a: real part

bi: imaginary part (where i = √ - 1)

### Trigonometric form

z: complex number

ρ: complex number module ()

Θ: argument z

(Moivre formula)

z: complex number

ρ: complex number module

n: exponent

Θ: argument z