50 Proven IB Math IA Topics for 2026

Below are 50 detailed topic ideas categorized by subject pathway and level. Each entry provides a refined research question alongside the primary mathematical tools required.

Category 1: IB Math AA HL Topics (Calculus, Proofs & Complex Algebra)

Math AA HL explorations require formal rigor, deep algebraic fluency, and sophisticated calculus techniques.

• 1. Projectile Motion with Quadratic Air Resistance: Modeling the trajectory of a basketball shot by setting up and solving first-order separable differential equations considering air resistance F_d = -kv^2.
• 2. Volume Optimization of Irregular Vessels: Using solids of revolution and definite integrals V = \pi \int_{a}^{b} [f(x)]^2 \, dx to calculate the precise capacity of a curved glass container and comparing it to experimental water displacement.
• 3. Fluid Dynamics & Torricelli’s Law: Deriving differential equations to model the drainage rate of liquid from conical vs. hemispherical funnels, validated with real video-tracking timing data.
• 4. Optimization via Lagrange Multipliers: Minimizing the material surface area required for a custom-designed ergonomic seating structure subject to a fixed volume constraint.
• 5. Proof & Applications of Euler’s Identity: Proving e^{i\pi} + 1 = 0 using Taylor Series expansion and exploring its geometrical representation via complex plane transformations.
• 6. Mandelbrot Set Boundary Dynamics: Investigating the convergence behavior of complex iterative functions z_{n+1} = z_n^2 + c and calculating approximate fractal dimensions.
• 7. 3D Vector Calculus in Satellite Signal Positioning: Calculating intersection coordinates of three directional vectors in 3D space to model GPS trilateration error margins.
• 8. The Brachistochrone Curve Problem: Applying the calculus of variations and parametric equations to prove that a cycloid represents the path of fastest descent under gravity.
• 9. Epidemic Outbreak Modeling via SIR Differential Equations: Constructing a system of non-linear differential equations \frac{dS}{dt}, \frac{dI}{dt}, \frac{dR}{dt} to predict peak infection rates during a seasonal flu outbreak.
• 10. Fourier Series & Harmonic Sound Decomposition: Decomposing complex musical instrument audio signals into infinite sums of sine and cosine functions.
• 11. Kepler’s Laws of Planetary Motion: Deriving elliptical orbital trajectories using polar coordinates and Newtonian gravitational calculus.
• 12. Newton’s Law of Cooling with Oscillating Ambient Temperature: Solving non-homogeneous differential equations where surrounding temperatures vary sinusoidally over time.

Category 2: IB Math AA SL Topics (Functions, Geometry & Single-Variable Calculus)

Math AA SL topics should focus on clear single-variable calculus, function transformations, and geometric optimization.

• 13. Sinusoidal Regression of Daylight Duration: Modeling annual daylight changes in Hong Kong vs. London using sine functions f(x) = a \sin(b(x-c)) + d and evaluating latitude effects.
• 14. Highway On-Ramp Curve Optimization: Applying first and second derivatives to determine optimal banking angles and curve radius to minimize centripetal vehicle slipping.
• 15. Logistic Growth Modeling of Endangered Species: Curve-fitting population growth data to a logistic function P(t) = \frac{L}{1 + e^{-k(t-t_0)}} to identify carrying capacity limits.
• 16. Topographical Mapping via Heron’s Formula: Calculating total irregular land parcel areas by sectioning terrain into non-right-angled triangles using satellite coordinate data.
• 17. The Golden Ratio in Modern Urban Architecture: Using logarithmic spirals and Fibonacci sequences to analyze structural aesthetic ratios in iconic Hong Kong skyscrapers.
• 18. Terminal Velocity of Falling Objects: Fitting exponential velocity curves to primary experimental drop data collected via high-speed phone camera tracking.
• 19. Harbor Tide Prediction for Maritime Navigation: Combining multiple sine functions to construct a accurate tidal prediction model for local container shipping lanes.
• 20. Catenary vs. Parabolic Bridge Cable Curves: Comparing hyperbolic cosine functions y = a \cosh(\frac{x}{a}) against standard quadratic polynomials in suspension bridge engineering.
• 21. Structural Efficiency of Honeycomb Hexagons: Proving geometrically why regular hexagons minimize perimeter-to-area ratios compared to triangles and squares.
• 22. Limits of Compound Interest: Evaluating financial accumulation as compounding frequency approaches infinity to derive \lim_{n \to \infty} (1 + \frac{1}{n})^n = e.
• 23. Bézier Curves in Vector Graphics: Utilizing cubic parametric equations to generate custom smooth typography curves in computer design software.
• 24. Damped Harmonic Motion of Simple Pendulums: Modeling mechanical energy loss over time using decay functions f(t) = A e^{-\gamma t} \cos(\omega t + \phi).

Category 3: IB Math AI HL Topics (Advanced Statistics, Graph Theory & Matrices)

Math AI HL students excel when utilizing multi-variable matrix operations, network algorithms, and inferential modeling.

• 25. Markov Chain Analysis of Board Game Dynamics: Constructing transition probability matrices and calculating steady-state eigenvectors to find winning probabilities in Monopoly.
• 26. Network Optimization via Dijkstra’s Algorithm: Mapping the Hong Kong MTR transit network as a weighted graph to calculate absolute shortest travel times between key hubs.
• 27. Google’s PageRank Matrix Algorithm: Building stochastic web link matrices and using power iteration methods to measure node authority scores.
• 28. Markowitz Portfolio Theory & Risk Minimization: Constructing variance-covariance matrices from stock return data to map an efficient investment frontier.
• 29. Multi-Factor Regression in Real Estate Valuation: Performing multiple linear regression with residual analysis and adjusted R^2 diagnostics to predict apartment prices.
• 30. Voting System Fairness & Arrow’s Theorem: Analyzing preference matrices across Borda Count, First-Past-The-Post, and Ranked Choice voting systems using historical election data.
• 31. Poisson Distribution in Sports Goal Modeling: Applying Poisson probability mass functions P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} to forecast European football match scorelines.
• 32. Travelling Salesman Problem for Logistics: Applying nearest-neighbor heuristics and graph theory to optimize multi-stop urban delivery routes.
• 33. Chi-Squared Independence Testing in Market Research: Conducting primary surveys and multi-way contingency table analysis to test consumer choices across demographics.
• 34. Game Theory & Oligopoly Price Strategy: Modeling competitive corporate pricing strategies using payoff matrices and Nash Equilibrium proofs.
• 35. Queueing Theory in Retail Service Lines: Utilizing Poisson arrival rates and exponential service times (M/M/1 queue models) to minimize customer wait times.
• 36. Voronoi Diagrams for Emergency Services Placement: Constructing spatial perpendicular bisector maps to optimize ambulance coverage zones across urban districts.
• 37. Time Series Forecasting of Currency Exchange Volatility: Utilizing autoregressive moving average models (ARIMA) to evaluate short-term exchange rate fluctuations.

Category 4: IB Math AI SL Topics (Applied Statistics & Real-World Modeling)

Math AI SL topics should leverage primary data collection, clear bivariate statistics, and accessible exponential modeling.

• 38. Caffeine Pharmacokinetics in the Human Body: Measuring personal cognitive reaction times post-coffee consumption and fitting exponential decay models C(t) = C_0 e^{-kt} to estimate half-life.
• 39. Sleep Duration vs. Academic Performance: Gathering primary student data to calculate Pearson’s correlation coefficient r and Spearman’s rank correlation between sleep hours and test scores.
• 40. Chi-Squared Goodness-of-Fit on Penalty Kicks: Analyzing professional football penalty kick directions to test whether shot placements follow a uniform distribution.
• 41. Newton’s Law of Cooling Experimental Verification: Collecting minute-by-minute temperature decay data of beverages in insulated vs. ceramic mugs to find cooling constants.
• 42. Depreciation Comparison of Electric vs. Petrol Vehicles: Curve-fitting 10-year secondary resale data using exponential decay vs. linear models.
• 43. Two-Sample t-Test on Reaction Times of Gamers: Conducting controlled experiments to test if action video game players exhibit statistically significant faster reaction times than non-gamers.
• 44. Probabilistic Breakdown of the Monty Hall Problem: Simulating thousands of trial rounds programmatically and using Bayes’ Theorem to prove why switching doors doubles win probability.
• 45. Logistic Modeling of Viral Content Dissemination: Tracking view counts of viral short videos over time and fitting sigmoidal growth curves.
• 46. Amortization Mechanics of Credit Card Debt: Modeling how minimum payment structures affect long-term interest accumulation using compound interest formulas.
• 47. Geometric Probability in Dartboard Aiming Strategies: Calculating target area probability ratios to determine optimal aiming coordinates for amateur dart throwers.
• 48. Acoustic Frequencies in Musical Tuning: Measuring audio frequencies across piano octaves using sound spectrum software to prove exponential frequency relationships f_n = f_0 \cdot 2^{n/12}.
• 49. Social Media Follower Retention Dynamics: Fitting logarithmic functions to measure user retention curves following major online content publishing events.
• 50. Standard Normal Distribution of Standardized Test Scores: Calculating Z-scores, standard deviations, and percentile rankings from public educational assessment data.