FREE FE Exam Study Guide 2026: The Shared Engineering Core, Built to the Exam

This free FE exam study guide teaches to the NCEES Fundamentals of Engineering exam — the first step toward becoming a licensed Professional Engineer.^[1] The FE comes in seven discipline versions, but they all rest on the same engineering core, so this guide teaches that broadly-shared core — math, probability and statistics, ethics, engineering economics, and the major engineering sciences — and shows you how to pick your discipline.

It is built to be deep and to actually teach: worked formulas, the high-yield rules each topic tests, and the relationships behind them — not a summary. And it’s interactive, with a checkpoint quiz after every module, hover-able glossary terms, and concept questions, so you learn by doing.

Read it module by module, test yourself at each checkpoint, then round out your free FE exam prep with our practice questions and flashcards.

FE Exam Snapshot

The FE rewards breadth — it samples a dozen-plus knowledge areas rather than going deep on a few. Because the exact mix depends on the discipline you choose, this guide focuses on the topics that show up across most disciplines. The shared core below is the biggest return on your study time:^[4]

Those are the NCEES question-count ranges for the FE Other Disciplines exam, the most general version.^[4] Other disciplines reshuffle the weights and add specialized areas (Civil adds surveying, structures, geotechnical, transportation; Mechanical adds machine design and controls), but the math, ethics, economics, and engineering-science fundamentals in this guide carry across all of them.

1 · Choosing Your Discipline & How the Exam Works

Before you study a single formula, get two things right: which FE you sit for and how the test is built. Both shape your prep.

The 7 FE Discipline Exams

NCEES offers sevenFE exams. You take the one that fits your degree or the license path you want — there is no “general” FE beyond Other Disciplines:^[1]

Format, Timing & Scoring

Every FE is 110 questions in a 6-hour appointment, delivered by computer (CBT) year-round at Pearson VUE test centers.^[1] The questions are mostly multiple choice, with some alternative item types (fill-in-the-blank, point-and-click, drag-and-drop). Here is how the clock actually breaks down:

Scoring is pass/fail. Your raw score (number correct — there is no penalty for wrong answers, so never leave a blank) is converted to a scaled score that adjusts for small differences in difficulty between exam forms, then compared to a minimum passing level set by subject-matter experts through a psychometric standard-setting process.^[2] There is no fixed percentage and no quota — first-time and repeat takers are held to the same standard, and failing examinees receive a diagnostic report by subject area.

The FE Reference Handbook

The FE is closed book except for one thing: the NCEES FE Reference Handbook, the only reference you may use. It appears on-screen as a searchable PDF during the exam, and the same version is free to download from your MyNCEES account.^[3] Treat it as a piece of equipment you must master, not a safety net.

Download the current handbook the day you start studying, and learn whereeach formula lives — searching for “Bernoulli” or “moment of inertia” under time pressure is slow if you have never done it. The handbook gives you the equations and tables but not which to use or how — that judgment is what this guide and your practice build.

Safety, Health & the Environment

Most FE exams test safety, health, and environmental awareness alongside ethics. The highest-yield piece is hazard communication: under the OSHA Hazard Communication Standard, containers of hazardous chemicals carry a label with a pictogram, signal word (Danger is more severe than Warning), and hazard statement, while the detailed information lives in the 16-section Safety Data Sheet (SDS).^[7]Know the difference between exposure limits, the precautionary principle, and an engineer’s duty to design for public and environmental safety.

2 · Mathematics

About 8–12 questions on Other Disciplines.^[4] Math underpins every other topic, so the payoff is larger than the count. The FE rewards recognizing what tool to apply, then pulling the formula from the handbook.

Calculus: Derivatives & Integrals

A derivative is an instantaneous rate of change (the slope of a curve); an integral is an accumulation (the area under a curve). The two are inverses by the fundamental theorem of calculus. Memorize the core derivatives and the rules that combine them:

Differential Equations

Classify first, solve second. A differential equation’s order is the highest derivative present; it is linear if the unknown and its derivatives appear only to the first power and are not multiplied together, and homogeneous when the right-hand side is zero. The single most common FE case is exponential growth/decay, $dP/dt = kP$, whose solution is $P = P_0 e^{kt}$ — positive k grows, negative decays.

Linear Algebra & Matrices

A system of linear equations has a unique solution when its coefficient matrix is square and its determinant is nonzero (the matrix is invertible). For a 2×2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$. Eigenvalues λ solve $\det(A - \lambda I) = 0$ and appear in vibration and stability problems.

Vectors & Analytic Geometry

A scalar has magnitude only; a vector has magnitude and direction. The dot product (A·B = |A||B|cos θ) gives a scalar and is zero for perpendicular vectors; the cross product (A×B = |A||B|sin θ) gives a vector perpendicular to both. These power force resolution in statics and dynamics.

3 · Probability & Statistics

About 6–9 questions.^[4] The FE tests interpreting data and computing simple probabilities — not heavy theory. Get the definitions exact and the rules straight.

Descriptive Statistics

The mean is the arithmetic average; the median is the middle ordered value and resists outliers; the mode is the most frequent value. Spread is measured by the variance (average squared deviation) and its square root, the standard deviation.

Probability Rules

For independent events, multiply: P(A and B) = P(A) × P(B). For mutually exclusive events, add: P(A or B) = P(A) + P(B). In general, P(A or B) = P(A) + P(B) − P(A and B), subtracting the double-counted overlap. Conditional probability is P(A | B) = P(A and B) / P(B).

Distributions & the Normal Curve

The normal distribution is the symmetric bell curve, fully described by its mean and standard deviation. By the empirical rule, about 68% / 95% / 99.7% of values fall within one, two, and three standard deviations of the mean.

A z-score = (x − mean) / standard deviation reports how many standard deviations a value sits from the mean. A confidence interval turns a sample estimate into a range likely to contain the true value.

4 · Engineering Ethics & Professional Practice

Roughly 4–8 questions depending on discipline — and among the most reliable points on the exam, because the answers follow a clear hierarchy.^[4] Master the rules and you bank these.

The Code of Ethics

Engineering codes of ethics (the NSPE Code is the standard reference) put one duty above all others: engineers must hold paramount the safety, health, and welfare of the public.^[6] When that duty conflicts with a client’s or employer’s wishes, the public comes first. Almost every ethics question reduces to applying that priority.

Duties, Conflicts & Confidentiality

Beyond public safety, engineers must: practice only in their area of competence; be objective and truthful in public statements; act as faithful agents for each client or employer; disclose conflicts of interest; and not accept gifts or bribes that could influence their judgment. Confidentiality is real but not absolute — a danger to public safety can require disclosure to the proper authority despite a nondisclosure agreement.

Licensure & Societal Impact

Licensure exists to protect the public by ensuring only qualified individuals offer engineering services. Engineers may seal or stamp only work done by them or under their responsible charge— stamping someone else’s work is both unethical and illegal. Modern codes also urge engineers to consider sustainability and societal impact, designing to protect the environment for future generations.

5 · Engineering Economics

About 4–9 questions depending on discipline.^[4] Engineering economics is high-yield because the method is the same every time: move money across time with an interest factor, then compare alternatives at a common point.

Time Value of Money

The time value of money is the whole subject in one idea: a dollar today is worth more than a dollar later because it can earn interest. Compounding moves money forward; discounting brings it back:

Interest Factors & Cash Flows

The handbook provides interest factor tables that convert between present worth (P), future worth (F), and a uniform annual series (A). An annuity is a series of equal payments at regular intervals; a gradient grows by a fixed amount each period. Sketch the cash-flow diagram first (arrows up for income, down for cost), then pick the factor that converts what you have into what you want.

Project Evaluation

To decide whether a project is worthwhile, bring all its cash flows to a common time and apply a rule. Net present value discounts everything to today — accept if it is positive. The internal rate of return is the rate that makes NPV zero — accept if it beats the minimum acceptable rate of return (MARR).

The benefit-cost ratio justifies a project when it is at least 1, and payback period is a quick (but time-value-blind) screen. Always ignore sunk costs.

6 · Statics

About 9–14 questions.^[4] Statics is the study of bodies in equilibrium — at rest or moving at constant velocity. Everything starts with a free-body diagram.

Equilibrium & Free-Body Diagrams

A body is in static equilibrium when forces and moments balance: ΣFx = 0, ΣFy = 0, ΣM = 0. The free-body diagram — the body isolated with every external force and reaction drawn — is what makes those equations solvable. The reactions depend on the support:

Trusses & Members

Truss members carry force only along their length (tension or compression). Solve them by the method of joints (apply ΣFx = 0 and ΣFy = 0 at each pin, starting where only two unknowns meet) or the method of sections (cut through the members of interest and apply equilibrium to one side). A two-force member loaded only at its two ends carries pure axial force; a zero-force member carries none.

Friction, Centroids & Moment of Inertia

Static friction resists impending motion up to a maximum of μₛN; once sliding starts, kinetic friction (usually a bit less) takes over. The centroidis the geometric center where a distributed load’s resultant acts, and the area moment of inertia describes resistance to bending — shifted between axes by the parallel-axis theorem ($I = I_c + Ad^2$).

7 · Dynamics

About 9–15 questions depending on discipline.^[4] Dynamics adds motion to statics. Separate the two halves cleanly: kinematics describes how things move; kinetics explains why.

Kinematics

Kinematics relates position, velocity, and acceleration. For constant acceleration, three equations do most of the work: v = v₀ + at, s = s₀ + v₀t + ½at², and v² = v₀² + 2a(s − s₀).

In projectile motion, the horizontal and vertical motions are independent — horizontal velocity is constant while vertical motion accelerates at g. Circular motion has a centripetal acceleration a = v² / r pointing toward the center.

Kinetics & Newton's Laws

Newton’s second law, F = ma, links the net force on a body to its acceleration. Its rotational analog is τ = Iα (torque equals mass moment of inertia times angular acceleration). Draw a free-body diagram, sum the forces, and solve for acceleration just as in statics — except the right-hand side is now ma, not zero.

Energy & Momentum Methods

Two shortcuts avoid solving for acceleration and time. The work-energy theorem (W = ΔKE) relates force-over-distance to a change in speed; when only conservative forces act, mechanical energy KE + PE is conserved. The impulse-momentum principle (FΔt = Δmv) handles collisions, where momentum is conserved if no external force acts.

8 · Mechanics of Materials

About 7–14 questions.^[4] Also called strength of materials, this topic asks how a body deforms and when it fails under load. It is heavily tested and rewards practice.

Stress, Strain & Hooke's Law

Normal stress is force per unit area perpendicular to a section ($\sigma = P/A$); strain is the fractional stretch ($\varepsilon = \Delta L / L$). Within the elastic region they obey Hooke’s law, $\sigma = E\varepsilon$, where E is the modulus of elasticity. The stress-strain curve tells the whole story:

Axial, Bending & Torsion

Three loading modes recur. Axial load stretches a bar by $\delta = PL/(AE)$. Bending produces a flexure stress $\sigma = Mc/I$, largest at the outer fiber and zero at the neutral axis. Torsion of a circular shaft produces shear stress $\tau = Tr/J$. Mohr’s circle converts a stress state into principal and maximum-shear stresses.

Columns & Factor of Safety

Slender columns fail by buckling — sudden sideways instability — at the critical load $P_{cr} = \pi^2 EI / (KL)^2$, where K depends on the end conditions. The factor of safety is the ratio of failure (or yield) strength to actual stress — the margin engineers build in to uphold public safety.

9 · Materials Science

About 5–11 questions depending on discipline.^[4]Materials science links a material’s internal structure to the properties an engineer cares about — strength, ductility, and durability.

Structure & Phase Diagrams

A phase diagram maps the stable phases of a material against temperature and composition; reading it predicts the microstructure that controls properties. In steels, higher carbon content raises hardness and strength but lowers ductility and weldability.

Heat treatments move a material between regions: annealing (slow cooling) softens and relieves stress, while quenching then tempering (fast cooling, then reheating) hardens while restoring some toughness. Smaller grains generally increase strength (the Hall-Petch relationship).

Mechanical Properties & Failure

Distinguish the property words the FE leans on: hardness (resistance to indentation), toughness (energy absorbed before fracture — the area under the stress-strain curve), ductility (ability to deform before breaking), and stiffness (the modulus E). Three time-dependent failure modes matter: fatigue (cracking under cyclic load below the static strength), creep (slow deformation under constant load at high temperature), and corrosion (electrochemical degradation, accelerated when dissimilar metals touch in an electrolyte — galvanic corrosion).

10 · Fluid Mechanics

About 6–18 questions — the largest single area on Other Disciplines.^[4] Fluid mechanics studies liquids and gases at rest and in motion. Two ideas — pressure and conservation — carry most of it.

Fluid Properties & Statics

Viscosityis a fluid’s resistance to shear; density and specific weight ($\gamma = \rho g$) describe how heavy it is. In a static fluid, pressure increases with depth: $P = \rho g h$. Pascal’s principle says a pressure change applied to an enclosed fluid transmits undiminished everywhere (the basis of hydraulics), and the buoyant force on a submerged body equals the weight of the fluid it displaces (Archimedes).

Continuity, Bernoulli & Flow

For flowing fluids, two equations dominate. The continuity equation conserves flow rate: A₁V₁ = A₂V₂, so a narrower pipe means faster flow. Bernoulli’s equation conserves energy along a streamline — pressure, kinetic, and potential energy trade off — so where velocity rises, pressure falls.

The Reynolds number ($Re = \rho V D / \mu$) tells you whether flow is laminar (Re below ~2300, smooth) or turbulent (above ~4000, mixing).

11 · Thermodynamics & Heat Transfer

About 9–14 questions (combined on Other Disciplines).^[4] Thermodynamics governs energy, heat, and work; heat transfer governs how thermal energy moves. The laws are the spine.

The Laws & Properties

The first law is energy conservation: $\Delta U = Q - W$ (heat added minus work done). The second law says total entropy never decreases — heat flows spontaneously from hot to cold, and no engine is perfectly efficient. For an ideal gas, PV = nRT ties pressure, volume, and absolute temperature together, and a temperature change needs heat $Q = mc\,\Delta T$.

Heat Engines & Efficiency

A heat engine converts heat into work while cycling between a hot and a cold reservoir. The best any engine can do is the Carnot efficiency, $\eta = 1 - T_{cold}/T_{hot}$, with temperatures in kelvin.

Heat Transfer

Thermal energy moves three ways — and the FE wants you to tell them apart and pick the right law:

12 · Electrical Engineering Fundamentals

About 5–9 questions on most non-electrical disciplines (and far more on FE Electrical and Computer).^[4] Even non-EE majors must handle basic circuits. Two laws cover most of it.

DC Circuits & the Basic Laws

Ohm’s law, V = IR, plus Kirchhoff’s laws (currents into a node sum to zero; voltages around a loop sum to zero) solve any resistive circuit. Resistors in series add (R = R₁ + R₂ + …); in parallel the reciprocals add (1/R = 1/R₁ + 1/R₂ + …), giving a total smaller than the smallest. Power is P = VI = I²R = V²/R.

AC Circuits & Reactance

In alternating current, capacitors and inductors add frequency-dependent reactance. Capacitive reactance falls with frequency (a capacitor blocks DC, passes high-frequency AC); inductive reactance rises with frequency (an inductor passes DC, opposes high-frequency AC).

Resistance and reactance combine into impedance ($Z = \sqrt{R^2 + X^2}$). The RMS value of a sinusoid is V_peak / √2, and the power factor is the cosine of the voltage-current phase angle.

How to Use This Study Guide

The FE is a breadth exam, so the winning strategy is steady coverage with focused practice — not one heroic cram. Most first-time takers pass; repeat takers pass far less often, so make your first attempt count.^[5]

FE Exam Glossary

Quick definitions for the terms you’ll see most across the FE engineering core:

Bernoulli's equation

Conservation of energy along a streamline for steady, incompressible, frictionless flow: pressure, kinetic, and potential energy trade off to a constant.

Buoyant force

The upward force on a submerged or floating body, equal to the weight of the fluid it displaces (Archimedes' principle).

Carnot efficiency

The maximum efficiency of a heat engine between two reservoirs, η = 1 − T_cold / T_hot, with temperatures in kelvin.

Centroid

The geometric center of an area or volume — the point about which the first moment of area is zero. The weight of a uniform body acts there.

Conduction, convection, radiation

The three modes of heat transfer: through a solid by contact, by a moving fluid, and by electromagnetic waves, respectively.

Confidence interval

A range computed from sample data that is expected to contain the true population parameter with a stated probability, such as 95%.

Continuity equation

For incompressible flow, A₁V₁ = A₂V₂: volumetric flow rate is conserved, so velocity rises where the area shrinks.

Engineering code of ethics

A professional society's rules (e.g., the NSPE Code) that require engineers to hold the public's safety, health, and welfare paramount.

Enthalpy

A property combining internal energy with flow work, H = U + PV; convenient for analyzing constant-pressure and flow processes.

Entropy

A property measuring a system's disorder or unavailable energy; it increases in any real (irreversible) process.

Euler buckling

The critical compressive load at which a slender column suddenly becomes unstable and bows out, dependent on length, stiffness, and end conditions.

Factor of safety

The ratio of a material's failure (or yield) strength to the actual applied stress; a margin against uncertainty in loads and properties.

Fatigue

Failure under repeated cyclic loading at stresses below the static strength, as cracks initiate and grow over many cycles.

First law of thermodynamics

Conservation of energy for a system: the change in internal energy equals heat added minus work done (ΔU = Q − W).

Free-body diagram

A sketch of a single body isolated from its surroundings, showing every external force and moment acting on it. It is the starting point for any statics problem.

Hooke's law

Within the elastic region, stress is proportional to strain, σ = Eε, where E is the modulus of elasticity (Young's modulus).

Impedance

The total opposition to alternating current, combining resistance and reactance: Z = √(R² + X²).

Impulse-momentum principle

The impulse of the net force (force times time) equals the change in momentum (FΔt = Δmv); momentum is conserved when no external force acts.

Internal rate of return (IRR)

The discount rate that makes a project's net present value equal to zero; compared to the minimum acceptable rate of return.

Kinematics vs kinetics

Kinematics describes motion — position, velocity, acceleration — without regard to forces; kinetics relates that motion to the forces causing it.

Kirchhoff's laws

KCL: currents into a node sum to zero (charge conserved). KVL: voltage rises and drops around a loop sum to zero (energy conserved).

Modulus of elasticity

Also Young's modulus (E): the slope of the elastic portion of the stress-strain curve, a measure of a material's stiffness.

Mohr's circle

A graphical method for transforming a known stress state to find the principal stresses and the maximum shear stress.

Moment of a force

The turning effect of a force about a point, equal to the force times the perpendicular distance from the point to the line of action (M = F × d).

Moment of inertia (area)

A geometric property of a cross-section describing its resistance to bending; larger when material is farther from the neutral axis.

Net present value (NPV)

The sum of a project's discounted cash flows minus its initial cost; a positive NPV means the project earns more than the required rate.

Newton's second law

The net force on a body equals its mass times its acceleration (F = ma); the foundation of dynamics and the link between force and motion.

Normal stress

Internal force per unit area acting perpendicular to a cross-section, σ = P / A, measured in pascals or psi.

Ohm's law

The relationship between voltage, current, and resistance in a circuit: V = I × R.

Phase diagram

A map showing the stable phases of a material as a function of temperature and composition; used to predict microstructure and properties.

Reynolds number

A dimensionless ratio of inertial to viscous forces (Re = ρVD / μ) that predicts whether flow is laminar or turbulent.

Second law of thermodynamics

The total entropy of an isolated system never decreases; heat flows spontaneously from hot to cold, limiting engine efficiency.

Shear stress

Internal force per unit area acting parallel to a surface, τ = V / A.

Static equilibrium

The condition in which all forces and all moments on a body sum to zero (ΣF = 0 and ΣM = 0), so the body neither accelerates nor rotates.

Strain

The fractional deformation of a material, ε = ΔL / L (dimensionless); normal strain stretches, shear strain distorts.

Time value of money

The principle that a sum today is worth more than the same sum later, because money now can earn a return; the basis of engineering economics.

Ultimate tensile strength

The maximum stress a material can withstand before it begins to fail; the peak of the stress-strain curve.

Viscosity

A fluid's resistance to shear or flow; the proportionality between shear stress and the velocity gradient.

Work-energy theorem

The net work done on a body equals its change in kinetic energy (W = ΔKE), letting you relate force over distance directly to a change in speed.

Yield strength

The stress at which a material begins to deform permanently (plastically) instead of returning to its original shape.

Free FE Exam Study Materials & Resources

Everything you need to prepare for the FE exam is free here — no paywall, no sign-up. This guide is the foundation; pair it with the rest of our free FE exam study materials for active recall and timed practice:

• FE Exam Practice Test — exam-style questions across the engineering core, with explanations.
• FE Exam Flashcards — active-recall decks for the high-yield formulas, laws, and definitions.

FE Exam Study Guide FAQ

References

1. 1.NCEES. “FE Exam — Fundamentals of Engineering.” NCEES. ↑
2. 2.NCEES. “Exam Scoring — How NCEES Scores the FE Exam.” NCEES. ↑
3. 3.NCEES. “NCEES Exam Reference Handbooks (FE Reference Handbook).” NCEES. ↑
4. 4.NCEES. “FE Other Disciplines — CBT Exam Specifications.” NCEES. ↑
5. 5.NCEES. “Squared 2024 — FE Exam Pass Rates (FY2024).” NCEES. ↑
6. 6.National Society of Professional Engineers. “NSPE Code of Ethics for Engineers.” NSPE. ↑
7. 7.U.S. Occupational Safety and Health Administration. “Hazard Communication Standard (29 CFR 1910.1200).” OSHA. ↑
8. 8.National Institute of Standards and Technology. “NIST/SEMATECH e-Handbook of Statistical Methods.” NIST. ↑
9. 9.National Institute of Standards and Technology. “The NIST Reference on Constants, Units, and Uncertainty.” NIST. ↑

Sources for the concept answers

Every answer in the FE exam concept questions above is drawn from an official primary source:

1. National Institute of Standards and Technology. “NIST Reference Fluid Properties.” NIST.
2. National Institute of Standards and Technology. “NIST Chemistry WebBook — Thermophysical Properties.” NIST.
3. National Institute of Standards and Technology. “NIST Reference on Thermodynamics.” NIST.
4. National Institute of Standards and Technology. “NIST Materials Data and Phase Diagram Resources.” NIST.

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