Complete Mathematics Formula Guide and Solver

Q: Can this solver solve equations with no real roots?

Yes, our solver fully supports complex numbers and displays solutions in standard a + bi form.

Q: Does the formula atlas solve every formula automatically?

Not yet. Interactive links are shown only for operations currently supported by the solver. The remaining entries are accurate reference material and define the roadmap for future solver modules.

Q: Can I link directly to one formula?

Yes. Each formula and topic has a stable fragment identifier, so the browser URL can point directly to that entry.

Overview & Core Concept

This reference organizes 64 essential formulas into ten sections spanning school mathematics, undergraduate foundations, and practical financial calculations. Search the atlas by name, symbol, or use case, then link directly to the exact formula you need.

A formula is useful only when its assumptions are clear. Entries therefore include applicability conditions where domain restrictions, units, invertibility, independence, or angle measures matter. Supported examples open directly in the Math Solver Suite with their values already filled from the URL.

The live solver currently covers common plane and solid geometry, quadratic and linear systems, 2x2 and 3x3 matrices, and selected derivatives and definite integrals. The wider atlas is a reference roadmap for expanding those interactive tools without pretending unsupported expressions can already be solved.

Step-by-Step: How to Use the Calculator

Search by Concept or Symbol

Use the formula search to narrow the full reference by a name such as Pythagoras, a symbol such as variance, or an application such as compound interest.

Check Conditions and Units

Read the usage condition before substituting values. Keep units consistent and verify restrictions such as non-zero denominators or radian angle measures.

Open a Worked Solver

Where an interactive implementation exists, use Solve an example to open the correct tab with a shareable query and prefilled values.

Formula reference

Complete Mathematics Formula Atlas

Search by concept, symbol, or use case. Every result has a permanent section link.

Find a formula

64 formulas in 10 sections

Arithmetic, Percentages, Sequences and Logarithms

Core rules for proportional reasoning, growth, sequences, indices, and logarithms.

Percentage Change

\%\text{ change}=\frac{\text{new}-\text{original}}{\text{original}}\times100

Measures a relative increase or decrease from an original value.

Use when: The original value is non-zero.

Direct and Inverse Proportion

y=kx \qquad y=\frac{k}{x}

Direct proportion keeps y/x constant; inverse proportion keeps xy constant.

Arithmetic Sequence and Sum

a_n=a_1+(n-1)d \qquad S_n=\frac{n}{2}\left(2a_1+(n-1)d\right)

Finds a term and the sum of the first n terms when consecutive terms differ by d.

Geometric Sequence and Sum

a_n=a_1r^{n-1} \qquad S_n=a_1\frac{1-r^n}{1-r} \qquad S_\infty=\frac{a_1}{1-r}

Models repeated multiplication by a common ratio r.

Use when: For the infinite sum, |r| < 1.

Laws of Indices

a^ma^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn},\quad a^{-n}=\frac{1}{a^n}

The standard exponent rules used to simplify powers.

Use when: Division and negative-index forms require a non-zero base.

Logarithm Laws

\log_b(xy)=\log_bx+\log_by,\quad \log_b(x^p)=p\log_bx,\quad \log_bx=\frac{\log_kx}{\log_kb}

Product, power, and change-of-base identities for logarithms.

Use when: x and y are positive; bases are positive and not equal to 1.

Algebra and Equations

Identities and solution formulas for linear, quadratic, polynomial, and exponential problems.

Linear Equation

ax+b=0 \Longrightarrow x=-\frac{b}{a}

Solves a one-variable equation in standard linear form.

Use when: a is non-zero.

Quadratic Formula

ax^2+bx+c=0 \Longrightarrow x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Finds both roots of any quadratic equation.

Use when: a is non-zero. The discriminant determines whether roots are real or complex.

Solve an example

Discriminant and Vertex

\Delta=b^2-4ac \qquad (h,k)=\left(-\frac{b}{2a},-\frac{\Delta}{4a}\right)

Classifies roots and locates the turning point of a parabola.

Essential Algebraic Identities

(a\pm b)^2=a^2\pm2ab+b^2,\quad a^2-b^2=(a-b)(a+b),\quad a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)

Common expansion and factorization identities.

Binomial Theorem

(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k,\qquad \binom{n}{k}=\frac{n!}{k!(n-k)!}

Expands a non-negative integer power of a binomial.

Exponential Growth and Decay

y=y_0e^{kt} \qquad t=\frac{1}{k}\ln\left(\frac{y}{y_0}\right)

Models continuously compounded growth when k > 0 and decay when k < 0.

Plane Geometry

Perimeter and area formulas for the most frequently used two-dimensional figures.

Rectangle and Square

A_{rectangle}=lw,\quad P_{rectangle}=2(l+w),\quad A_{square}=s^2,\quad P_{square}=4s

Area and perimeter from side lengths.

Solve an example

Triangle Area

A=\frac{1}{2}bh \qquad A=\sqrt{p(p-a)(p-b)(p-c)},\quad p=\frac{a+b+c}{2}

Uses base and perpendicular height, or Heron's formula when all three sides are known.

Solve an example

Pythagorean Theorem

a^2+b^2=c^2

Relates the legs a and b to hypotenuse c in a right triangle.

Use when: The angle opposite c is 90 degrees.

Circle

A=\pi r^2,\quad C=2\pi r,\quad L_{arc}=r\theta,\quad A_{sector}=\frac{1}{2}r^2\theta

Area, circumference, arc length, and sector area.

Use when: Use radians for the arc and sector formulas.

Solve an example

Parallelogram and Trapezoid

A_{parallelogram}=bh \qquad A_{trapezoid}=\frac{1}{2}(a+b)h

Areas from perpendicular height and parallel side lengths.

Regular Polygon

A=\frac{1}{2}ap=\frac{ns^2}{4\tan(\pi/n)},\qquad \theta_{interior}=\frac{(n-2)180^\circ}{n}

Area and each interior angle of a regular n-sided polygon.

Solid Geometry and Mensuration

Surface area and volume formulas for common three-dimensional solids.

Cube and Cuboid

V_{cube}=s^3,\quad S_{cube}=6s^2,\qquad V_{cuboid}=lwh,\quad S_{cuboid}=2(lw+lh+wh)

Volume and total surface area from edge dimensions.

Cylinder

V=\pi r^2h,\qquad S=2\pi r(r+h)

Volume and total surface area of a right circular cylinder.

Solve an example

Cone

\ell=\sqrt{r^2+h^2},\quad V=\frac{1}{3}\pi r^2h,\quad S=\pi r(r+\ell)

Slant height, volume, and total surface area of a right circular cone.

Solve an example

Sphere

V=\frac{4}{3}\pi r^3,\qquad S=4\pi r^2

Volume and surface area from radius.

Solve an example

Prism and Pyramid

V_{prism}=A_bh,\qquad V_{pyramid}=\frac{1}{3}A_bh

General volume rules using base area A_b and perpendicular height h.

Conical Frustum

V=\frac{\pi h}{3}(R^2+Rr+r^2),\qquad \ell=\sqrt{h^2+(R-r)^2}

Volume and slant height of a cone cut parallel to its base.

Trigonometry

Ratios and identities for triangles, periodic functions, and angle transformations.

Right-Triangle Ratios

\sin\theta=\frac{opposite}{hypotenuse},\quad \cos\theta=\frac{adjacent}{hypotenuse},\quad \tan\theta=\frac{opposite}{adjacent}

Defines sine, cosine, and tangent for an acute angle in a right triangle.

Pythagorean Identities

\sin^2\theta+\cos^2\theta=1,\quad 1+\tan^2\theta=\sec^2\theta,\quad 1+\cot^2\theta=\csc^2\theta

Fundamental identities used to transform and solve trigonometric expressions.

Angle Addition and Subtraction

\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\quad \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B

Expands sine and cosine of a sum or difference of angles.

Double-Angle Identities

\sin2A=2\sin A\cos A,\quad \cos2A=\cos^2A-\sin^2A=1-2\sin^2A=2\cos^2A-1

Expresses functions of twice an angle using functions of the original angle.

Sine Rule

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R

Relates each side of a triangle to its opposite angle and circumradius R.

Cosine Rule

c^2=a^2+b^2-2ab\cos C

Generalizes the Pythagorean theorem to any triangle.

Triangle Area with Included Angle

A=\frac{1}{2}ab\sin C

Finds triangle area from two sides and their included angle.

Coordinate Geometry and Vectors

Formulas for lines, distances, conic sections, and vector operations in two and three dimensions.

Distance and Midpoint

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\qquad M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Finds the length and midpoint of a line segment in the Cartesian plane.

Slope and Equation of a Line

m=\frac{y_2-y_1}{x_2-x_1},\qquad y-y_1=m(x-x_1),\qquad y=mx+c

Equivalent forms for the gradient and equation of a non-vertical straight line.

Point-to-Line Distance

d=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}

Shortest distance from point (x_0,y_0) to line Ax + By + C = 0.

Equation of a Circle

(x-h)^2+(y-k)^2=r^2

Represents a circle with center (h,k) and radius r.

Vector Magnitude and Unit Vector

|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2},\qquad \hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}

Computes vector length and a direction vector of length one.

Use when: The zero vector has no unit vector.

Dot Product and Angle

\mathbf{a}\cdot\mathbf{b}=\sum_i a_ib_i=|\mathbf{a}||\mathbf{b}|\cos\theta

Tests perpendicularity, computes projections, and finds the angle between vectors.

Cross Product

\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix},\qquad |\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin\theta

Produces a vector perpendicular to two 3D vectors; its magnitude is parallelogram area.

Calculus

Limits, differentiation, integration, and series formulas used across science and engineering.

Derivative from First Principles

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}

Defines instantaneous rate of change as the limit of secant slopes.

Product, Quotient, and Chain Rules

(uv)'=u'v+uv',\quad \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2},\quad \frac{d}{dx}f(g(x))=f'(g(x))g'(x)

The principal rules for differentiating combined and composite functions.

Standard Derivatives

\frac{d}{dx}x^n=nx^{n-1},\quad \frac{d}{dx}e^x=e^x,\quad \frac{d}{dx}\ln x=\frac1x,\quad (\sin x)'=\cos x,\quad (\cos x)'=-\sin x

The most-used elementary derivative formulas.

Solve an example

Fundamental Theorem of Calculus

\int_a^b f(x)\,dx=F(b)-F(a),\qquad F'(x)=f(x)

Connects definite integration with antiderivatives.

Standard Integrals

\int x^n dx=\frac{x^{n+1}}{n+1}+C,\quad \int e^x dx=e^x+C,\quad \int\frac1x dx=\ln|x|+C,\quad \int\sin x dx=-\cos x+C

Common antiderivatives used as building blocks.

Use when: For the power rule, n is not -1.

Solve an example

Substitution and Integration by Parts

\int f(g(x))g'(x)dx=\int f(u)du,\qquad \int u\,dv=uv-\int v\,du

Two foundational transformations for evaluating non-basic integrals.

Taylor and Maclaurin Series

f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n,\qquad e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}

Approximates a sufficiently differentiable function near a chosen center a.

Matrices and Linear Algebra

Matrix operations, determinants, inverses, linear systems, and eigenvalue relationships.

Matrix Product

(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}

Computes each result entry as a row-by-column dot product.

Use when: The number of columns of A equals the number of rows of B.

2 x 2 Determinant

\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc

Measures signed area scaling and determines invertibility.

2 x 2 Matrix Inverse

\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}

Produces the matrix that reverses multiplication by A.

Use when: ad - bc is non-zero.

Cramer's Rule

x_i=\frac{\det(A_i)}{\det(A)}

Solves a square linear system by replacing one coefficient column at a time.

Use when: The coefficient matrix A is square and det(A) is non-zero.

Eigenvalues and Eigenvectors

A\mathbf{v}=\lambda\mathbf{v},\qquad \det(A-\lambda I)=0

Finds directions preserved by a linear transformation and their scale factors.

Rank-Nullity Theorem

\operatorname{rank}(A)+\operatorname{nullity}(A)=n

Splits the dimension of the domain into the row-space contribution and null space.

Probability, Combinatorics and Statistics

Counting rules, conditional probability, distributions, and descriptive statistics.

Permutations and Combinations

{}^nP_r=\frac{n!}{(n-r)!},\qquad {}^nC_r=\binom nr=\frac{n!}{r!(n-r)!}

Counts ordered arrangements and unordered selections without replacement.

Addition and Multiplication Rules

P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A\cap B)=P(A)P(B|A)

Combines probabilities for unions and intersections of events.

Conditional Probability and Bayes' Theorem

P(A|B)=\frac{P(A\cap B)}{P(B)},\qquad P(A|B)=\frac{P(B|A)P(A)}{P(B)}

Updates the probability of A after observing evidence B.

Use when: P(B) is non-zero.

Mean, Variance, and Standard Deviation

\bar{x}=\frac1n\sum_{i=1}^n x_i,\qquad \sigma^2=\frac1N\sum_{i=1}^N(x_i-\mu)^2,\qquad \sigma=\sqrt{\sigma^2}

Summarizes center and spread for sample or population data.

Expected Value and Variance

E[X]=\sum_x xP(X=x),\qquad \operatorname{Var}(X)=E[X^2]-E[X]^2

Computes the long-run mean and spread of a discrete random variable.

Binomial Distribution

P(X=k)=\binom nkp^k(1-p)^{n-k},\qquad E[X]=np,\qquad \operatorname{Var}(X)=np(1-p)

Models the number of successes in n independent trials.

Use when: Trials have two outcomes and a constant success probability p.

Standard Score

z=\frac{x-\mu}{\sigma}

Expresses an observation as its number of standard deviations from the mean.

Financial Mathematics

Interest, annuity, loan, and investment formulas used in personal finance.

Simple Interest

I=Prt,\qquad A=P(1+rt)

Calculates interest only on the original principal.

Compound Interest

A=P\left(1+\frac{r}{n}\right)^{nt},\qquad A=Pe^{rt}

Future value under periodic or continuous compounding.

Present Value

PV=\frac{FV}{(1+r)^n}

Discounts a future amount back to its value today.

Future Value of an Annuity

FV=P\frac{(1+i)^n-1}{i} \qquad FV_{due}=P\frac{(1+i)^n-1}{i}(1+i)

Accumulated value of equal end-of-period or beginning-of-period payments.

Loan Payment or EMI

M=P\frac{i(1+i)^n}{(1+i)^n-1}

Fixed periodic payment required to amortize a loan.

Compound Annual Growth Rate

CAGR=\left(\frac{V_f}{V_i}\right)^{1/n}-1

Equivalent constant annual growth rate between initial and final values.

Why Our Visual Calculator Outperforms Generic Tools

Market Comparison Matrix

Move from reference to calculation without losing the original problem or its shareable URL.

Evaluation Metric	Generic Market Calculators	RV Anveshana Visual Suite
Formula navigation	Long static page	Search and stable section links
Worked calculations	Separate calculator pages	Prefilled solver links
Mathematical typography	Varies by source	KaTeX HTML and MathML

Formula navigation

Competitors: Long static page

RV Anveshana: Search and stable section links

Worked calculations

Competitors: Separate calculator pages

RV Anveshana: Prefilled solver links

Mathematical typography

Competitors: Varies by source

RV Anveshana: KaTeX HTML and MathML

Generic Tool Limitations

Static formula sheets separate reference material from calculation tools.
Unstructured formula lists are difficult to scan, search, and deep-link.
A final numeric answer often omits the assumptions behind the chosen formula.

Our Visual Superpowers

Searchable sections with stable links to every formula.
Direct transitions from supported formulas to prefilled solver examples.
Accessible KaTeX output with both HTML and MathML representations.

Practical Case Study & Real Numbers

Case Study: Solving 2x² - 4x + 5 = 0

Scenario: A high school student encounters an equation with negative discriminant.

Numerical Breakdown

a = 2, b = -4, c = 5. Discriminant Δ = (-4)² - 4(2)(5) = 16 - 40 = -24 (negative). Roots = [4 ± √(-24)] / 4 = [4 ± 2i√6] / 4 = 1 ± (i√6)/2.

Key Takeaway: Our solver displays both complex roots clearly formatted with imaginary i units and radical simplification.

Pro Tips & Critical Pitfalls

Expert Pro Tips

Identify the required output and known quantities before choosing a formula; similar-looking formulas often assume different inputs.
Keep units consistent throughout a calculation and convert only once, preferably before substitution.
For trigonometry and calculus, confirm whether angles are measured in degrees or radians.
Estimate the expected magnitude first so a misplaced sign, exponent, or decimal is easier to detect.

Common Pitfalls

Using a formula without checking its domain restrictions or geometric assumptions.
Rounding intermediate values too early instead of rounding only the final answer.
Treating correlation as causation or using dependent-event formulas as if events were independent.
Forgetting to distribute negative signs when substituting signed coefficients.

Frequently Asked Questions

Yes, our solver fully supports complex numbers and displays solutions in standard a + bi form.

Not yet. Interactive links are shown only for operations currently supported by the solver. The remaining entries are accurate reference material and define the roadmap for future solver modules.

Yes. Each formula and topic has a stable fragment identifier, so the browser URL can point directly to that entry.

Ready to Run Your Exact Numbers?

Launch Academic Math Solver Suite now to dynamically adjust inputs with instant dual-sliders, inspect schedules, and simulate your scenarios.

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Overview & Core Concept

Step-by-Step: How to Use the Calculator

Search by Concept or Symbol

Check Conditions and Units

Open a Worked Solver

Complete Mathematics Formula Atlas

Percentage Change

Direct and Inverse Proportion

Arithmetic Sequence and Sum

Geometric Sequence and Sum

Laws of Indices

Logarithm Laws

Linear Equation

Quadratic Formula

Discriminant and Vertex

Essential Algebraic Identities

Binomial Theorem

Exponential Growth and Decay

Rectangle and Square

Triangle Area

Pythagorean Theorem

Circle

Parallelogram and Trapezoid

Regular Polygon

Cube and Cuboid

Cylinder

Cone

Sphere

Prism and Pyramid

Conical Frustum

Right-Triangle Ratios

Pythagorean Identities

Angle Addition and Subtraction

Double-Angle Identities

Sine Rule

Cosine Rule

Triangle Area with Included Angle

Distance and Midpoint

Slope and Equation of a Line

Point-to-Line Distance

Equation of a Circle

Vector Magnitude and Unit Vector

Dot Product and Angle

Cross Product

Derivative from First Principles

Product, Quotient, and Chain Rules

Standard Derivatives

Fundamental Theorem of Calculus

Standard Integrals

Substitution and Integration by Parts

Taylor and Maclaurin Series

Matrix Product

2 x 2 Determinant

2 x 2 Matrix Inverse

Cramer's Rule

Eigenvalues and Eigenvectors

Rank-Nullity Theorem

Permutations and Combinations

Addition and Multiplication Rules

Conditional Probability and Bayes' Theorem

Mean, Variance, and Standard Deviation

Expected Value and Variance

Binomial Distribution

Standard Score

Simple Interest

Compound Interest

Present Value

Future Value of an Annuity

Loan Payment or EMI

Compound Annual Growth Rate

Why Our Visual Calculator Outperforms Generic Tools

Practical Case Study & Real Numbers

Case Study: Solving 2x² - 4x + 5 = 0

Pro Tips & Critical Pitfalls

Frequently Asked Questions

Can this solver solve equations with no real roots?

Does the formula atlas solve every formula automatically?

Can I link directly to one formula?

Ready to Run Your Exact Numbers?

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