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Formula Explorer

Discover, visualize, and understand math formulas

CBSE Class:
Calculus
Class 11-12

Derivative & Tangent Line

f(x) = ax², f'(x₀) = 2ax₀

The derivative gives the slope (rate of change) at any point. The tangent line touches the curve at exactly one point and has the slope equal to the derivative there.

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Algebra
Class 9-10

Absolute Value

y = a|x - h| + k

A V-shaped graph. The vertex is at (h, k). When a is negative, it flips upside down into a mountain shape.

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Algebra
Class 11-12

Exponential Growth

y = a * b^x

Exponential growth — starts slow then explodes upward. When b > 1 it grows, when 0 < b < 1 it decays.

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Algebra
Class 11-12

Reciprocal / Hyperbola

y = a / (x - h) + k

A hyperbola with two separate branches. Has vertical asymptote at x = h and horizontal asymptote at y = k.

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Calculus
Class 11-12

Simple Harmonic Motion

x = A cos(w*t + phi)

The motion of a pendulum or spring — back and forth at a constant rate. A is how far it swings, w is how fast.

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Physics
Class 10-12

Lens Formula (Optics)

1/f = 1/v - 1/u

Relates focal length (f), image distance (v), and object distance (u) for thin lenses. Determines where an image forms.

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Engineering
Engineering

Stress-Strain (Engineering)

σ = E * ε (Stress = Young's Modulus × Strain)

Hooke's law for materials: stress is proportional to strain in the elastic region. E (Young's modulus) measures material stiffness.

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Engineering
Engineering

Beam Deflection

δ = F*L³ / (3*E*I)

Maximum deflection of a cantilever beam under a point load at the free end. E is Young's modulus, I is moment of inertia.

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Engineering
Class 11+

Torque

τ = r * F * sin(θ)

Rotational force. Depends on the force magnitude, distance from the pivot, and the angle of application.

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Finance
Advanced

Moving Average (Trading)

SMA = Average of last n prices

Simple Moving Average smooths price data by averaging the last n periods. Used to identify trends and support/resistance levels.

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Finance
Advanced

Sharpe Ratio

Sharpe = (Portfolio Return - Risk-Free Rate) / Portfolio Std Dev

Measures risk-adjusted return. A higher Sharpe ratio means better return per unit of risk taken.

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Computer Science
Engineering

Big-O Complexity

Comparing algorithm growth rates

How algorithm runtime scales with input size. O(n log n) is the sweet spot for sorting; O(n²) becomes impractical for large data.

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AI & Data Science
Class 11-12

Bayes' Theorem

P(A|B) = P(B|A) × P(A) / P(B)

Bayes' Theorem updates your belief about something when you get new evidence. If a medical test is 95% accurate and a disease affects 1% of people, a positive result does NOT mean 95% chance of being sick — it is actually closer to 16%. Bayes explains why, and it is the foundation of all modern AI.

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AI & Data Science
Class 12+

Cross-Entropy Loss

L = -sum(y × log(y_hat))

Cross-entropy loss measures how wrong an AI model is prediction is. If the model says 90% chance of cat and it IS a cat, the loss is low (0.1). If the model says 90% cat but it is a dog, the loss is high (2.3). The AI learns by minimizing this number across all training examples.

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Daily Life
Class 6-8

Recipe Scaling

New quantity = Original × (new servings / original servings)

Scaling a recipe is pure ratio math. If a recipe for 4 people needs 2 cups of flour, then for 6 people you need 2 × 6/4 = 3 cups. Simple proportional reasoning — but it gets tricky with cooking times (which don not scale linearly) and seasoning (which scales sub-linearly).

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Mathematics
Class 12+

Euler's Identity

e^(i*pi) + 1 = 0

Called 'the most beautiful equation in mathematics,' Euler's Identity connects five fundamental constants: e (growth), i (imaginary), pi (circles), 1 (unity), and 0 (nothing). It shows that exponential growth, imaginary numbers, and circles are all secretly the same thing. The general form e^(ix) = cos(x) + i·sin(x) connects exponentials to rotation.

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Economics
Class 8-10

Simple & Compound Interest Comparison

SI = P×r×t, CI = P(1+r)^t - P

Simple interest grows linearly — like adding the same amount every year. Compound interest grows exponentially — interest earns interest. Over long periods, the difference is staggering. 1 lakh at 10% for 30 years: SI gives 3 lakhs, CI gives 17.4 lakhs — nearly 6x more!

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Finance
Advanced

Value at Risk (VaR)

VaR_α = Percentile(Gains/Losses, 100 - α)

Maximum potential loss of a portfolio over a given time period at a specified confidence level. At 95% confidence, VaR is the 5th percentile of the gain/loss distribution — meaning there is only a 5% chance of losing more than this amount.

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AI & Data Science
Class 9+

Summation (Sigma Notation)

Sum from i=1 to n of f(i)

Sigma notation is the compact way to write "add up a bunch of things." Every AI loss function, every training step, every gradient update uses summation. When you see MSE = (1/n)Sum(errors), that Sum IS sigma notation. Without it, writing AI math would take pages instead of one line.

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AI & Data Science
Class 10+

Mean Squared Error (MSE)

MSE = (1/n) * Sum of (actual - predicted)^2

MSE is the most fundamental loss function in machine learning. It measures how wrong your predictions are by squaring each error (so big mistakes are punished much more than small ones) and averaging them. A perfect model has MSE = 0. Every regression model from linear regression to neural networks can use MSE as its training objective.

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AI & Data Science
Class 10+

Cosine Similarity

cos(θ) = (a · b) / (||a|| * ||b||)

Cosine similarity measures the angle between two vectors, ignoring their magnitude. It outputs a value between -1 (opposite) and +1 (identical direction). This makes it perfect for comparing things where magnitude does not matter — like comparing document topics regardless of document length, or comparing user preferences regardless of how active they are.

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AI & Data Science
Class 11+

Chain Rule (Calculus)

d/dx f(g(x)) = f'(g(x)) * g'(x)

The chain rule is the mathematical engine behind backpropagation — the algorithm that trains every neural network. It tells you how to compute the derivative of a composed function by multiplying derivatives step by step. Without the chain rule, we could not train deep networks because we would have no way to calculate how changing one weight affects the final output through multiple layers.

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AI & Data Science
Advanced

Scaled Dot-Product Attention

Attention(Q,K,V) = softmax(Q*K^T / sqrt(d_k)) * V

The attention mechanism is the core innovation behind Transformers and all modern LLMs. It computes how much each element in a sequence should "attend to" every other element. Q (query) asks "what am I looking for?", K (key) says "what do I contain?", and V (value) says "what information do I provide." The dot product QK^T measures relevance, scaling by sqrt(d_k) prevents gradient issues, softmax creates a probability distribution, and multiplying by V retrieves the relevant information.

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Algebra
Class 10

Arithmetic Progression - Sum of n Terms

Sₙ = (n/2)[2a + (n-1)d]

Sum of first n terms of an AP. Also written as Sₙ = n/2(a + l) where l is the last term.

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Algebra
Class 11

Geometric Progression - Sum of n Terms

Sₙ = a(rⁿ-1)/(r-1)

Sum of first n terms of a GP. For |r| < 1, the sum approaches a/(1-r) as n→∞.

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Algebra
Class 11

Binomial Theorem

(a+b)ⁿ = Σ C(n,k) × a^(n-k) × b^k

Expansion of (a+b)ⁿ into n+1 terms using binomial coefficients. Pascal's triangle gives the coefficients.

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Trigonometry
Class 10

Trigonometric Identity - Pythagorean

sin²θ + cos²θ = 1

The fundamental trigonometric identity derived from the Pythagorean theorem. Forms the basis for all other trig identities.

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Trigonometry
Class 11

Compound Angle - Cosine

cos(A+B) = cosA×cosB - sinA×sinB

Cosine of sum of two angles. Note the minus sign — it's the opposite of the sine formula.

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Trigonometry
Class 11

Double Angle Formula - Sine

sin(2θ) = 2×sinθ×cosθ

Sine of double angle. Derived from sin(A+B) with A=B. Useful for solving equations and simplifying expressions.

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Trigonometry
Class 11-12

Sine Wave

y = A sin(wx + phi)

An oscillating wave. Amplitude (A) controls height, frequency (w) controls how fast it oscillates, and phase (phi) shifts it left/right.

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Geometry
Class 9-10

Circle Equation

(x-h)² + (y-k)² = r²

A circle with center at point (h, k) and radius r. All points on the circle are exactly r units away from the center.

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Statistics
Class 11-12

Normal Distribution (Bell Curve)

Bell curve: f(x) = (1/sigma*sqrt(2*pi)) * e^(-(x-mu)²/(2*sigma²))

The bell curve shows probability distribution. Most values cluster near the mean (mu). Sigma controls the spread: smaller sigma = taller & narrower peak.

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Calculus
3D
Class 12+

3D Paraboloid

z = a(x² + y²)

A 3D paraboloid surface. The coefficient a controls how steep the bowl shape is. Positive a opens upward, negative opens downward.

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Trigonometry
Class 11-12

Cosine Wave

y = A cos(wx + phi)

Like sine but starts at its peak. Cosine and sine are identical waves, just shifted by 90 degrees.

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Trigonometry
Class 11-12

Tangent Function

y = a tan(bx)

An ever-increasing wave that shoots to infinity at regular intervals. Has vertical asymptotes where it is undefined.

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Geometry
Class 11-12

Ellipse

(x-h)²/a² + (y-k)²/b² = 1

An oval shape — a stretched circle. Semi-axis a goes horizontal, b goes vertical. When a = b, it becomes a circle.

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Geometry
Class 9-10

Area of Triangle (Heron)

Area = sqrt(s(s-a)(s-b)(s-c)), s = (a+b+c)/2

Calculates the area of any triangle when you know all three sides. s is the semi-perimeter (half of total perimeter).

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Calculus
Class 12+

Logistic Growth

y = L / (1 + e^(-k*(x-x0)))

S-shaped curve that starts slow, accelerates, then levels off at a maximum L. Models population growth with limited resources.

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Physics
Class 10-12

Snell's Law (Refraction)

n₁ sin(θ₁) = n₂ sin(θ₂)

When light passes from one medium to another, it bends. This law tells you exactly how much it bends based on the refractive indices.

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Physics
Class 9-11

Kinetic Energy

KE = 0.5 * m * v²

Energy of a moving object. Doubling the speed quadruples the energy — which is why high-speed crashes are so devastating.

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Physics
Class 9-11

Newton's Law of Gravitation

F = G * m₁ * m₂ / r²

Every object attracts every other object with a force proportional to their masses and inversely proportional to the square of the distance between them.

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Physics
Class 11

Simple Pendulum Period

T = 2π * sqrt(L / g)

The time for one complete swing depends only on the pendulum length and gravity — not on mass or amplitude (for small angles).

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Physics
Class 11-12

Wave Equation

y = A sin(k*x - w*t)

A traveling wave: k is the wave number (2π/wavelength), ω is angular frequency (2π/period). The wave moves in the +x direction.

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Physics
Class 11-12

Doppler Effect

f' = f * (v + vo) / (v - vs)

The observed frequency changes when source or observer moves. Approaching = higher pitch; receding = lower pitch.

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Physics
Class 12+

Stefan-Boltzmann Law

P = σ * A * T⁴ (σ = 5.67×10⁻⁸)

The total power radiated by a black body is proportional to the fourth power of its temperature. Double the temperature = 16x the radiation.

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Finance
Advanced

Black-Scholes Option Pricing

C = S*N(d1) - K*e^(-r*T)*N(d2)

The Nobel Prize-winning formula for pricing European call options. It considers stock price, strike price, time, volatility, and risk-free rate.

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Space
Class 11+

Orbital Velocity

v = sqrt(G*M / r)

The speed needed to maintain a circular orbit at radius r. Higher orbits = slower speed.

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Chemistry
Class 11-12

Ideal Gas Law

P*V = n*R*T

Relates pressure, volume, temperature, and amount of gas. Increasing temperature at constant volume increases pressure.

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Computer Science
Engineering

Neural Network Activation (ReLU)

f(x) = max(0, x)

Rectified Linear Unit — the most popular activation function in deep learning. Simple: outputs zero for negative inputs, passes positive inputs through.

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Mathematics
Class 8-12

Fibonacci Sequence

F(n) = F(n-1) + F(n-2), F(0)=0, F(1)=1

The Fibonacci sequence starts with 0 and 1, and every next number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... This deceptively simple pattern appears everywhere in nature, art, architecture, finance, and computer science. The ratio of consecutive Fibonacci numbers converges to the Golden Ratio (phi = 1.618...), one of the most beautiful constants in mathematics.

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Medicine
Class 8-10

Body Mass Index (BMI)

BMI = weight (kg) / height (m)²

BMI estimates body fat based on weight and height. Under 18.5 is underweight, 18.5-24.9 is normal, 25-29.9 is overweight, and 30+ is obese. Used worldwide by doctors for quick health screening.

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Medicine
Class 11-12

Drug Half-Life

C(t) = C₀ × (1/2)^(t/t_half)

After taking medicine, the drug concentration in your blood decreases over time. The half-life is how long it takes for half the drug to be eliminated. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. This determines how often you need to take your medicine.

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AI & Data Science
Class 12+

Gradient Descent

theta_new = theta - learning_rate × gradient

Gradient descent is how AI learns. Imagine you are blindfolded on a hilly landscape and want to reach the lowest valley. You feel the slope under your feet and take a step downhill. Repeat. That is gradient descent — the algorithm adjusts weights step by step to minimize error. Every neural network in the world learns this way.

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Finance
Advanced

Geometric Brownian Motion (GBM)

daily_return = exp[(μ - 0.5σ²)Δt + σ√Δt × Z]

Models how asset prices evolve randomly over time. Combines a deterministic drift (μ) with random shocks (σ × Z). The foundation of modern option pricing and Monte Carlo simulations in quantitative finance.

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Finance
Advanced

Merton Model (Credit Risk)

PD = N(-d2), d2 = [ln(V₀/D) + (r - 0.5σ²)T] / (σ√T)

Estimates the probability of a firm defaulting on its debt by modeling the firm's asset value as a stochastic process. Default occurs when asset value falls below the face value of debt at maturity. Uses the Black-Scholes framework applied to corporate credit risk.

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AI & Data Science
Class 10+

Dot Product

a · b = Sum(a_i * b_i) = ||a|| * ||b|| * cos(θ)

The dot product measures how similar two vectors are in direction. It is the single most important operation in all of AI — every neural network layer, every attention head, every embedding lookup is fundamentally a dot product. When two vectors point the same way, the dot product is large and positive. When perpendicular, it is zero. When opposite, it is negative.

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Algebra
Class 10

Sum and Product of Roots

Sum = -b/a, Product = c/a

For quadratic ax² + bx + c = 0, the sum of roots equals -b/a and product equals c/a. Useful for forming equations when roots are known.

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Algebra
Class 10

Arithmetic Progression - nth Term

aₙ = a + (n-1)d

The nth term of an AP where a is first term and d is common difference. Every term differs from the previous by exactly d.

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Trigonometry
Class 11

Compound Angle - Sine

sin(A+B) = sinA×cosB + cosA×sinB

Sine of sum of two angles. Essential for deriving double angle formulas and solving trigonometric equations.

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Algebra
Class 10-12

Quadratic Equation

y = ax² + bx + c

A parabola (U-shape curve). The value of a controls whether it opens up or down and how wide it is. b shifts the vertex left/right, and c moves it up/down.

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Geometry
Class 7-9

Pythagorean Theorem

a² + b² = c²

In a right triangle, the square of the hypotenuse (longest side) equals the sum of squares of the other two sides. This is one of the most fundamental theorems in mathematics!

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Linear Algebra
Graduation

2D Matrix Transformation

[x', y'] = R(angle) * S(scale) * [x, y]

Matrices can rotate, scale, and transform shapes. Watch how a unit square transforms as you change the rotation angle and scaling factors.

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Statistics
Class 11-12

Linear Regression

y_hat = intercept + slope * x

Linear regression finds the best straight line through scattered data points. It minimizes the total distance between the line and all data points.

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Calculus
3D
Class 12+

3D Saddle Surface

z = a(x² - y²)

A saddle (hyperbolic paraboloid) curves up in one direction and down in another, like a horse saddle or a Pringles chip.

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Trigonometry
3D
Class 12+

3D Wave Surface

z = A sin(wx * x) cos(wy * y)

A 3D wave surface combining sine and cosine waves. Adjust amplitude and frequencies to see beautiful wave interference patterns.

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Trigonometry
Class 12+

Damped Oscillation

y = A * e^(-d*x) * sin(w*x)

A wave that gradually fades away. The damping factor d controls how quickly the oscillations die out.

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Statistics
Class 12+

Sigmoid / Logistic Curve

sigma(x) = 1 / (1 + e^(-k*x))

An S-shaped curve between 0 and 1. Used as activation function in neural networks and for probability modeling.

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Physics
Class 9-11

Newton's Second Law

F = m * a

Force equals mass times acceleration. The heavier an object and the faster you want it to accelerate, the more force you need.

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Physics
Class 11-12

Projectile Motion

y = x*tan(θ) - g*x²/(2*v₀²*cos²(θ))

The path of an object launched at an angle — a parabola. The launch angle and speed determine how far and how high it goes.

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Physics
Class 12

RC Circuit Charging

Vc = V₀ * (1 - e^(-t/(R*C)))

A capacitor charges exponentially through a resistor. The time constant τ = RC determines how fast it charges.

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Physics
Class 12

Bohr Model Energy Levels

Eₙ = -13.6 / n² eV

Electron energy in a hydrogen atom. Energy is quantized — only specific levels are allowed. n=1 is the ground state.

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Engineering
Engineering

Gear Ratio

GR = N_driven / N_driver

Ratio of teeth between two meshing gears. Higher ratio means more torque but less speed at the output.

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Finance
Class 8+

Compound Interest

A = P * (1 + r/n)^(n*t)

Money grows exponentially when interest is reinvested. The more frequently interest compounds, the faster it grows.

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Space
Class 11+

Kepler's Third Law

T² = (4π² / GM) * a³

The square of orbital period is proportional to the cube of semi-major axis. Farther planets orbit slower.

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Space
Advanced

Hubble's Law

v = H₀ * d

Galaxies recede from us at speeds proportional to their distance. This proves the universe is expanding.

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Medicine
Class 12+

Epidemic SIR Model

dI/dt = beta × S × I - gamma × I

The SIR model divides a population into Susceptible, Infected, and Recovered. Beta is how contagious the disease is, gamma is the recovery rate. The ratio R₀ = beta/gamma determines if an epidemic spreads (R₀ > 1) or dies out (R₀ < 1). This is the model that guided COVID-19 policy worldwide.

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Economics
Class 11-12

Supply-Demand Equilibrium

Supply: Qs = a + bP, Demand: Qd = c - dP, Equilibrium: a+bP = c-dP

When supply equals demand, you get the equilibrium price — the natural market price. If price is too high, supply exceeds demand (surplus) and price falls. Too low, demand exceeds supply (shortage) and price rises. This is the most fundamental law of economics.

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Mathematics
Class 11-12

Permutations & Combinations

C(n,r) = n! / (r! × (n-r)!)

How many ways can you choose r items from n items? If order matters (like passwords), it is permutations. If order does not matter (like teams), it is combinations. C(52,5) = 2,598,960 possible poker hands. C(49,6) = 13,983,816 possible lottery combinations. These numbers explain why winning the lottery is nearly impossible.

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Finance
Advanced

Expected Shortfall (CVaR)

ES_α = Mean of losses beyond VaR threshold

Average loss in the worst-case scenarios beyond the VaR threshold. While VaR answers "how bad could it get?", Expected Shortfall answers "when it gets bad, how bad on average?" — capturing tail risk severity that VaR ignores.

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AI & Data Science
Class 11+

Softmax Function

softmax(z_i) = e^z_i / Sum(e^z_j)

Softmax converts any set of raw numbers (logits) into a valid probability distribution that sums to 1. It is the final layer of virtually every classification neural network, and it is how ChatGPT decides which word to say next. The exponential function ensures all outputs are positive, and the division normalizes them into probabilities.

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AI & Data Science
Advanced

KL Divergence

D_KL(P||Q) = Sum of P(x) * log(P(x)/Q(x))

KL Divergence measures how different one probability distribution is from another. It is asymmetric — D(P||Q) is not equal to D(Q||P). In AI, it measures how well a model distribution Q approximates the true distribution P. It is zero only when the distributions are identical. KL Divergence is the core of variational inference, VAEs, and is hidden inside every cross-entropy loss.

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Algebra
Class 11

Geometric Progression - nth Term

aₙ = a × r^(n-1)

The nth term of a GP where a is first term and r is common ratio. Each term is multiplied by r to get the next.

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Algebra
Class 9-11

Logarithm Laws

log(xy) = log(x) + log(y)

Logarithms convert multiplication to addition. Other laws: log(x/y) = log(x) - log(y), log(xⁿ) = n×log(x).

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Algebra
Class 8-10

Linear Equation

y = mx + b

A straight line where m is the slope (steepness) and b is the y-intercept (where the line crosses the y-axis). Change m to tilt the line; change b to shift it up or down.

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Calculus
Class 12

Integration (Area Under Curve)

Area = integral of f(x) from a to b

Integration calculates the total area under a curve between two points. It is the reverse of differentiation and represents accumulation.

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Algebra
Class 11-12

Cubic Function

y = ax³ + bx

A cubic curve with an S-shape. It can have one hump and one dip. Coefficient a controls the stretch and direction.

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Algebra
Class 11-12

Logarithmic Function

y = a * log(x) / log(b)

The inverse of exponential — it grows quickly at first then slows down. Only defined for positive x values.

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Trigonometry
Class 12+

Lissajous Curve

x = sin(a*t), y = sin(b*t + phi)

Beautiful curves made by combining two perpendicular oscillations. The ratio a:b determines the pattern shape.

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Geometry
Class 11

Parabola (Conic)

y² = 4ax (or x = t², y = 2at)

A parabola opening to the right. The focus is at (a, 0). This is the conic section form, different from y = x².

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Calculus
Class 12

Exponential Decay

y = A * e^(-k*x)

Starts high and falls toward zero, never quite reaching it. Models cooling, radioactive decay, and depreciation.

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Statistics
Class 11-12

Polynomial Regression

y = a*x² + b*x + c + noise

Fitting a curved line through scattered data points. Better than linear regression when data has a curve.

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Statistics
Class 12

Binomial Distribution

P(X=k) = C(n,k) * p^k * (1-p)^(n-k)

Probability of exactly k successes in n trials. Like flipping a coin n times and counting heads.

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Physics
Class 12

Coulomb's Law

F = k * q₁ * q₂ / r²

The electric force between two charges. It gets dramatically stronger as charges get closer (inverse square law).

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Physics
Class 10-12

Ohm's Law

V = I * R

Voltage equals current times resistance. More resistance means less current flows for the same voltage.

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Physics
Class 11

Hooke's Law (Spring Force)

F = -k * x

A spring exerts a restoring force proportional to its displacement. The negative sign means the force opposes the displacement.

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Engineering
Engineering

Bernoulli's Equation

P + 0.5*ρ*v² + ρ*g*h = constant

For fluid flow: as velocity increases, pressure decreases. This explains airplane lift, garden hose nozzles, and much more.

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Engineering
Engineering

Heat Transfer (Conduction)

Q = k * A * (Th - Tc) / L

Rate of heat flow through a material. Depends on thermal conductivity (k), area (A), temperature difference, and thickness (L).

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Finance
Class 10+

SIP Returns (Annuity)

FV = P * ((1+r)^n - 1) / r

Future value of a Systematic Investment Plan. Regular monthly investments compound over time to build wealth.

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Space
Class 11+

Escape Velocity

vₑ = sqrt(2*G*M / r)

Minimum speed to escape a planet's gravity without further propulsion. For Earth, it's about 11.2 km/s.

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Chemistry
Class 10-12

pH Scale

pH = -log10([H+])

Measures acidity/basicity. pH 7 is neutral, below 7 is acidic, above 7 is basic. Each unit is a 10x change in H+ concentration.

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Chemistry
Class 12

Arrhenius Equation

k = A * e^(-Ea / (R*T))

Reaction rate increases exponentially with temperature. Ea is activation energy — the energy barrier for the reaction.

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Computer Science
Engineering

Shannon Entropy

H = -Σ pᵢ log₂(pᵢ)

Measures information content and uncertainty. Maximum entropy when all outcomes are equally likely; zero when the outcome is certain.

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Economics
Class 11-12

EMI Calculator

EMI = P × r(1+r)^n / ((1+r)^n - 1)

EMI (Equated Monthly Installment) is the fixed amount you pay every month for a loan. It includes both principal repayment and interest. In early months, most of the EMI goes toward interest; later, more goes toward principal. This formula is used for home loans, car loans, and education loans.

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Daily Life
Class 8-10

Calorie Burn Rate

Calories = MET × weight(kg) × time(hours)

MET (Metabolic Equivalent of Task) measures how many calories an activity burns compared to resting. Sitting = 1 MET, walking = 3.5 MET, running = 8-12 MET, sleeping = 0.9 MET. Multiply MET by your weight and time to get calories burned.

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Daily Life
Class 6-8

Percentage & Percentage Change

% change = (new - old) / old × 100

Percentage is the most used math concept in daily life. A 20% discount on a 1000 rupee shirt saves you 200. A 10% salary hike on 50,000 gives you 5,000 more. Understanding percentages means you can never be fooled by misleading statistics, sales tricks, or financial jargon.

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Daily Life
Class 5-8

Speed, Distance & Time

D = S × T

The most basic formula in motion. If you drive at 60 km/h for 2 hours, you cover 120 km. Rearrange it: Time = Distance/Speed (how long to get there?) or Speed = Distance/Time (how fast were you going?). Every navigation app in the world uses this formula as its foundation.

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Algebra
Class 10

Quadratic Formula

x = (-b ± √(b²-4ac)) / 2a

The universal solution for any quadratic equation ax² + bx + c = 0. The discriminant (b²-4ac) tells you whether roots are real or imaginary.

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Trigonometry
Class 11

Sine Rule

a/sinA = b/sinB = c/sinC = 2R

In any triangle, side lengths are proportional to sines of opposite angles. R is the circumradius.

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Calculus
Class 11-12

Derivative from First Principles

f'(x) = lim[h→0] (f(x+h) - f(x))/h

The fundamental definition of derivative as instantaneous rate of change. This limit is what makes calculus possible.

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Calculus
Class 12

Basic Integration - Power Rule

∫xⁿdx = x^(n+1)/(n+1) + C

Reverse of differentiation. The constant C accounts for any constant term lost during differentiation.

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Calculus
Class 12

Area Under Curve

Area = ∫[a to b]|f(x)|dx

Geometric interpretation of definite integral. Absolute value ensures area is always positive.

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Statistics
Class 11

Standard Deviation

σ = √[Σ(xᵢ-x̄)²/n]

Measures spread of data around the mean. Low σ = data clustered near mean; high σ = data spread out.

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Statistics
Class 12

Binomial Probability Distribution

P(X=r) = C(n,r) × pʳ × (1-p)^(n-r)

Probability of exactly r successes in n independent trials with success probability p.

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Linear Algebra
Class 12

Matrix Addition

[A+B]ᵢⱼ = aᵢⱼ + bᵢⱼ

Matrices of same dimensions are added element by element. Order doesn't matter (commutative).

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Linear Algebra
Class 12

Matrix Multiplication

[AB]ᵢⱼ = Σₖ aᵢₖ × bₖⱼ

Row of first matrix × column of second matrix. A(m×n) × B(n×p) = C(m×p). Not commutative: AB ≠ BA generally.

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Linear Algebra
Class 12

Vector Dot Product

a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃

Scalar product measures alignment of vectors. Perpendicular vectors have dot product = 0.

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Linear Algebra
Class 12

Vector Cross Product

a × b = |a||b|sin(θ) n̂

Vector product gives perpendicular vector. Magnitude equals parallelogram area. Direction by right-hand rule.

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Geometry
Class 9-10

Circumference of Circle

C = 2πr

Perimeter of a circle. The ratio C/d = π was one of the first mathematical discoveries.

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Geometry
Class 9-10

Volume of Sphere

V = (4/3)πr³

Space enclosed by a sphere. Volume grows as cube of radius — double radius = 8× volume.

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Geometry
Class 9-10

Volume of Cylinder

V = πr²h

Volume equals base area times height. Circular cross-section extended through height.

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Geometry
Class 9-10

Pythagoras Theorem

c² = a² + b²

In a right triangle, square of hypotenuse equals sum of squares of other two sides. Perhaps the most famous theorem.

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Geometry
Class 10

Section Formula

((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Coordinates of point dividing line segment joining (x₁,y₁) and (x₂,y₂) in ratio m:n internally.

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Geometry
Class 10

Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Special case of section formula with m:n = 1:1. The midpoint divides the segment into two equal parts.

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Geometry
Class 10-11

Slope of a Line

m = (y₂-y₁)/(x₂-x₁) = tan(θ)

Slope measures steepness of a line. Positive slope = rising line; negative = falling; zero = horizontal; undefined = vertical.

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Geometry
Class 10-11

Equation of Line - Slope-Intercept

y = mx + c

Standard form of line equation where m is slope and c is y-intercept. Most intuitive form for graphing.

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Geometry
Class 11-12

Parabola Standard Form

y² = 4ax

Standard parabola with vertex at origin, focus at (a,0). All points are equidistant from focus and directrix.

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Calculus
Class 12

Chain Rule

d/dx[f(g(x))] = f'(g(x)) × g'(x)

For composite functions, multiply the outer derivative by the inner derivative. Essential for differentiating nested functions.

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Statistics
Class 12

Bayes Theorem

P(A|B) = P(B|A) × P(A) / P(B)

Updates probability based on new evidence. The foundation of Bayesian statistics and modern AI.

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Trigonometry
Class 11

Cosine Rule

c² = a² + b² - 2ab×cosC

Generalization of Pythagoras theorem for any triangle. Reduces to a² + b² = c² when C = 90°.

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Geometry
Class 11

Standard Circle Equation

(x-h)² + (y-k)² = r²

Circle with center (h,k) and radius r. All points at distance r from center satisfy this equation.

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Geometry
Class 10

Tangent Length from External Point

L = √(d² - r²)

Length of tangent from external point at distance d from center to circle of radius r. Uses Pythagoras theorem.

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Geometry
Class 11-12

Ellipse Standard Form

x²/a² + y²/b² = 1

Ellipse with semi-major axis a and semi-minor axis b. Sum of distances from any point to both foci is constant (2a).

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Geometry
Class 11-12

Hyperbola Standard Form

x²/a² - y²/b² = 1

Hyperbola with transverse axis 2a. Difference of distances from any point to both foci is constant (±2a).

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Calculus
Class 12

Product Rule

d/dx[f×g] = f'g + fg'

Derivative of a product of two functions. "First times derivative of second plus second times derivative of first."

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Calculus
Class 12

Integration - Trigonometric

∫sin(x)dx = -cos(x) + C

Integral of sine is negative cosine. Similarly, ∫cos(x)dx = sin(x) + C.

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Calculus
Class 12

Integration - Exponential

∫eˣdx = eˣ + C

The exponential function is its own integral (and derivative). Unique among all functions.

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Statistics
Class 9-10

Mean (Arithmetic Average)

x̄ = Σxᵢ/n

Sum of all values divided by count. The most common measure of central tendency.

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Statistics
Class 10

Probability - Classical Definition

P(E) = Favorable outcomes / Total outcomes

Probability of event E is ratio of favorable outcomes to total possible outcomes, assuming equally likely outcomes.

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Statistics
Class 12

Conditional Probability

P(A|B) = P(A∩B) / P(B)

Probability of A given that B has occurred. Fundamental to Bayesian reasoning and machine learning.

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Linear Algebra
Class 12

Determinant (2×2 Matrix)

det(A) = ad - bc

Determinant measures "scaling factor" of linear transformation. Zero determinant means transformation collapses dimensions.

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Linear Algebra
Class 12

Cramer's Rule

x = det(Aₓ)/det(A), y = det(Aᵧ)/det(A)

Solves linear equations using determinants. Replace column with constants to get numerator determinant.

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Trigonometry
Class 11

Double Angle Formula - Cosine

cos(2θ) = cos²θ - sin²θ

Cosine of double angle with three equivalent forms. Choose the form based on what's given in the problem.

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Trigonometry
Class 11-12

Sum to Product - Sine

sinA + sinB = 2×sin((A+B)/2)×cos((A-B)/2)

Converts sum of sines into a product. Essential for solving equations and analyzing wave interference.

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Trigonometry
Class 12

Inverse Trigonometric Functions

sin⁻¹(x) + cos⁻¹(x) = π/2

Fundamental relationship between inverse sine and cosine. Other identities: tan⁻¹x + cot⁻¹x = π/2.

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Geometry
Class 10

Distance Formula

d = √[(x₂-x₁)² + (y₂-y₁)²]

Distance between two points in 2D space. Direct application of Pythagoras theorem to coordinate geometry.

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Calculus
Class 11

Power Rule

d/dx(xⁿ) = n×x^(n-1)

The most fundamental differentiation rule. Works for any real exponent n, including fractions and negatives.

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Calculus
Class 12

Quotient Rule

d/dx[f/g] = (f'g - fg')/g²

Derivative of a quotient. "Low d-high minus high d-low, over low squared."

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Calculus
Class 12

Maxima and Minima Conditions

f'(x)=0, f''(x)<0 → maximum; f''(x)>0 → minimum

Critical points occur where derivative is zero. Second derivative test determines if it's a maximum or minimum.

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Calculus
Class 12

Definite Integral

∫[a to b]f(x)dx = F(b) - F(a)

The Fundamental Theorem of Calculus. Definite integral equals antiderivative evaluated at bounds.

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Geometry
Class 9-10

Area of Circle

A = π × r²

Area enclosed by a circle of radius r. Derived by integrating 2πr from 0 to r.

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Geometry
Class 9-10

Surface Area of Sphere

A = 4πr²

Surface area of a sphere equals 4 times the area of a circle with same radius.

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Geometry
Class 9-10

Volume of Cone

V = (1/3)πr²h

Cone volume is exactly one-third of a cylinder with same base and height.

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Geometry
Class 9

Heron's Formula

A = √[s(s-a)(s-b)(s-c)], s = (a+b+c)/2

Triangle area from three sides. s is semi-perimeter. Works when height is unknown or hard to find.

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