Free Prompt Library

Universal AI Prompt Toolkit

Useful prompts for clearer thinking, faster work, better decisions, stronger writing, and more honest answers. These can be used with ChatGPT or nearly any conversational AI.

Need math help? Jump to the formulas ↓

Objective analysis & perspective

For accuracy, resistance, verification, and a view outside your own assumptions.

/truth — Give me the honest answer

Separate evidence from confidence and challenge weak assumptions.

Do not tell me what you think I want to hear. Give me the most accurate answer you can. Separate verified facts, reasonable inferences, opinions, and unknowns. Challenge my assumptions when they appear wrong. If evidence is missing, say so plainly. Do not manufacture certainty, praise, or reassurance.

/bottomline — Lead with the answer

Get the conclusion before the background.

Give me the bottom line first. Then explain only the facts and reasoning needed to support it. Do not bury the answer beneath background information, disclaimers, or unnecessary detail.

/clarify — Find the real question

Resolve missing information without starting an interview.

Before answering, identify anything missing that would materially change the result. Ask no more than three focused questions. If the missing information would not significantly change the answer, state your assumptions and proceed.

/devil — Test the opposite position

Expose flaws by arguing the strongest other side.

Act as a devil's advocate. Build the strongest reasonable case against my current idea or position and identify three major flaws or blind spots I may be missing. Do not use weak objections or exaggeration. Then explain which objections are valid, which are manageable, and whether the original position still holds.

/bias — Remove the slant

Identify emotional, political, cultural, or ideological framing.

Review this material for political, cultural, ideological, or emotional bias. Identify loaded assumptions or wording, then rewrite it in a neutral, factual, and balanced tone without hiding meaningful disagreements.

/audit — Find what is wrong or missing

Review a plan, document, process, or system as an auditor.

Review this as an auditor, not a cheerleader. Look for errors, contradictions, missing requirements, broken links, weak assumptions, hidden costs, legal or financial exposure, security or safety concerns, and failure points. Rank findings by importance and do not invent problems merely to fill the list.

/stress-test — Try to break the idea

Test against real users, bad data, delays, and failures.

Assume this plan will face real users, mistakes, delays, bad data, unexpected costs, and partial failures. Identify where it is most likely to break, the consequences, and the smallest practical changes that would make it stronger.

/verify — Check before agreeing

Independently test names, dates, calculations, and claims.

Independently verify the names, dates, calculations, assumptions, and claims in what I provide. Do not assume my information is correct. Show any mismatch, explain why it matters, and state what remains uncertain. If current information or external sources are required and you can access them, use them.

/source-check — Separate evidence from claims

Label claims as sourced, inferred, or unverified.

For every important claim, identify whether it is supported by a source, inferred from available evidence, or currently unverified. Prefer primary sources when available. Include direct links if your tools allow browsing. Do not present search-result summaries as proof.

/no-invention — Stay inside the evidence

Prevent invented details from entering the answer.

Use only the information I supplied or information you can verify. Do not invent names, dates, features, quotes, files, events, motivations, or completed work. If something is missing, label it unknown.

/sanity-check — Does this make sense?

Check realism, consistency, and proportionality.

Check whether this idea is internally consistent, realistic, and proportionate to the problem. Identify the strongest part, weakest part, any hidden assumption, and whether I should proceed, revise, or stop.

High-speed productivity

For reducing reading, isolating action, and turning unstructured material into useful work.

/short — Keep it tight

Remove excess without losing what is necessary.

Answer as concisely as possible without omitting information necessary to understand or act on the answer. No long introduction, repetition, motivational language, or unnecessary formatting.

/tldr — Condense it fast

Turn long material into a rapid status view.

Summarize the material below into three punchy bullet points under 10 words each. Preserve the most important decision, risk, and next action.

/action — Extract the work

Pull tasks, owners, and deadlines from messy notes.

Extract only the actionable tasks, deadlines, assigned owners, dependencies, and unresolved decisions from this material. Put them into a clean, prioritized list and preserve exact dates and names.

/draft — Write the reply

Create a usable message with the right boundary and tone.

Draft a clear, professional reply to the message below. Preserve my intended meaning and level of directness. State the decision plainly, maintain the relationship where possible, and do not add promises I did not authorize.

/agenda — Structure the meeting

Convert a broad goal into a timed working session.

Create a structured, minute-by-minute agenda for a 45-minute meeting focused on solving this problem: [insert issue]. Include the desired decision, necessary inputs, time limits, assigned follow-up, and a clear definition of done.

/extract — Pull out what matters

Organize important information from any source.

Extract the important information from this material. Organize it into facts, decisions, tasks, dates, people, costs, links, open questions, and risks. Preserve exact numbers and wording where accuracy matters.

/rewrite — Improve without changing my voice

Clean up writing without making it generic or corporate.

Rewrite this for clarity, grammar, and flow while preserving my meaning, personality, level of directness, and intended audience. Do not make it sound corporate, overly polite, or unlike me.

Knowledge & education

For understanding a subject, teaching it, or turning passive reading into active learning.

/plain — Use normal language

Translate technical material without deleting important facts.

Explain this in direct, everyday language. Avoid jargon unless necessary, and define any technical term you must use. Start with what it means in practice, then explain how it works and what can go wrong.

/deep — Examine it thoroughly

Request full context, tradeoffs, edge cases, and consequences.

Analyze this in depth. Include relevant context, dependencies, tradeoffs, edge cases, risks, and likely consequences. Distinguish important issues from minor ones and finish with a clear conclusion.

/eli5 — Explain it simply

Build an intuitive foundation with a concrete analogy.

Explain the concept below as though I am five years old. Use one simple, concrete real-world analogy first, then add the minimum extra detail needed to keep the explanation accurate.

/8020 — Find the useful core

Learn the small set of concepts that produces most practical understanding.

I want to learn [insert topic]. Identify the 20% of core concepts that will yield roughly 80% of the practical understanding. Put them in learning order and give me one small exercise for each.

/quiz — Test my recall

Turn reading material into an active study tool.

Turn the material below into a five-question multiple-choice quiz. Test understanding rather than trivia. Put the answer key and a one-sentence explanation for each answer at the very bottom.

/socratic — Guide me without answering

Learn by working through the problem yourself.

Do not give me the answer. Ask me one focused guiding question at a time to help me solve this problem myself. Use each answer I give to decide the next question.

Problem solving & decisions

For diagnosing roadblocks, comparing realistic choices, and converting decisions into action.

/coach — Diagnose the roadblock

Find the root cause before prescribing a fix.

Act as an expert consultant. Ask me one diagnostic question at a time to uncover why this project or task is failing. Once the likely root cause is supported, propose the smallest practical fix and a test that would prove it worked.

/optimize — Remove waste

Improve a process, system, workflow, or code.

Review this process, system, workflow, or code. Identify redundancy, bottlenecks, unnecessary complexity, and wasted resources. Propose a simpler version that improves speed and reliability without removing required safeguards.

/unstick — Create new starting points

Break a creative or practical deadlock.

I am stuck on [insert task]. Give me five genuinely different angles, hooks, or approaches. Make at least one safe, one unconventional, and one immediately testable. Recommend the best place to restart.

/compare — Make the tradeoffs visible

Evaluate all options using the same criteria.

Compare the available options using the same criteria. Include cost, time, difficulty, risk, maintenance, reversibility, and likely outcome where relevant. Recommend one option and explain why it wins for my stated goal.

/decision — Help me choose

Turn possibilities into a ranked recommendation.

Identify the actual decision, constraints that matter, and consequences of each realistic option. Recommend the best choice rather than leaving me with an unranked list. State what new information could change the recommendation.

/three-options — Give me useful choices

See a safe, balanced, and ambitious route.

Give me three genuinely different approaches: 1. the safest or simplest option; 2. the balanced option; 3. the most ambitious option. Explain the main benefit and tradeoff of each, then recommend one.

/plan — Turn it into action

Build an ordered path with dependencies and checkpoints.

Convert this goal into an ordered, practical plan. Identify prerequisites, dependencies, checkpoints, likely blockers, and the definition of done. Put the next physical or digital action first.

/one-step — Work one step at a time

Avoid overload and verify progress before continuing.

Give me only the next step. Make it specific and testable. Wait for my result before giving the following step. Do not repeat completed checks unless new evidence makes them necessary.

/next-stone — Keep the work moving

Find the next unfinished action that materially advances the goal.

Review what has already been completed and identify the next unfinished action that materially advances the goal. Do not recheck settled work without a concrete reason. Bring the next paving stone with you.

/checkmath — Audit the numbers

Recalculate quantities, rates, fees, and totals independently.

Recalculate everything independently from the original quantities and rates. Show the formula, intermediate values, rounding, units, and final total. Identify taxes, fees, or assumptions that may have been counted twice or omitted.

/recover — Resume an off-track task

Return to the last verified good state without blindly restarting.

Determine the last verified good state, what changed afterward, and the first failing boundary. Do not restart everything blindly. Propose the smallest reversible action that can restore progress, along with a test that proves whether it worked.

Continuity, presentation & completion

For preserving work, transferring it cleanly, improving its presentation, and verifying completion.

/checkpoint — Capture where we are

Preserve the state of a long or complicated effort.

Create a concise checkpoint containing: the goal, completed work, verified results, decisions already made, current blocker, files or links or values that matter, and the exact next step. Do not add speculative history or omit unresolved problems.

/handoff — Prepare this for another AI

Make work understandable without the current conversation.

Create a self-contained handoff for another AI that has no access to this conversation. Include the objective, relevant background, completed work, verified evidence, decisions, constraints, unresolved issues, important files or identifiers, and the exact next action. Clearly separate facts from assumptions.

/first-look — See it like a new user

Expose confusion hidden by your private knowledge.

Evaluate this as though you are encountering it for the first time with no private context. Explain what you believe it is, who it is for, what it offers, what it costs, what you are expected to do next, and anything that creates confusion or distrust.

/sell-it — Make the value clear

Present an offer persuasively without dishonest pressure.

Present this persuasively without exaggeration, unsupported claims, fake urgency, or corporate filler. Lead with the user's problem, show how this solves it, explain the practical value, address the main objection, and end with a clear next action.

/public-safe — Remove private information

Prepare internal material for public sharing.

Create a version safe for public sharing. Remove or generalize credentials, private URLs, personal identifiers, internal architecture, customer information, security details, and anything that would help someone misuse the system. Preserve enough substance for the material to remain useful.

/done-check — Prove it is finished

Compare the result with the request and definition of done.

Compare the result against the original request and definition of done. List what is verified complete, what is only claimed or assumed, what remains unfinished, and the exact evidence needed before declaring success.

Flight Command Tips & Tricks

Make continuity a working habit.

Everything below this line is specific to Flight Command: active projects, real Workshop files, continuity, visible approvals, backups, and work that carries forward between conversations.

The shortest useful FCA routine

Flight Command works best when you establish a few simple habits with the AI you use every day. The words can be anything that feels natural to you; what matters is agreeing on what they mean.

  1. Establish a signal system with your AI.

    Select Flight Command from Plugins and agree with your everyday AI on simple phrases for beginning and ending work. Set them up while working through Flight Command so your AI can follow the same routine when you return.

    “Good morning” can mean: connect to Flight Command, confirm the active project, review the latest continuity, and tell me where we stopped.

    “Good night” or “Back up this chat” can mean: review this conversation and preserve the important progress, decisions, files, unresolved questions, and next steps in the appropriate Workshop project.

    These are not required Flight Command commands. They are personal signals agreed upon between you and your AI. Choose words that fit the way you naturally work.

  2. Confirm the correct project. Make sure the conversation is connected to the work you intend to continue or change.
  3. Continue from the real work. Retrieve established project information instead of asking AI to recreate it from conversational memory.
  4. Leave the project ready to continue. Before you go, preserve the important result, settled decisions, current status, and exact next step.

Ten habits that make FCA more useful

01

Give each real job its own project.

Keep unrelated work separated so each project's files, decisions, and history remain together.

02

Use names you will recognize later.

Name projects, folders, and files for what they contain—not merely when they were created.

03

Settle important decisions in writing.

Record what was decided, why it was chosen, and what it replaces.

04

Save continuity at natural stopping points.

Preserve the state after a breakthrough, before changing conversations, or when the next step becomes clear.

05

Keep source material with the work.

Store documents, images, code, notes, specifications, and references inside the project.

06

Ask ChatGPT to retrieve before it assumes.

Use established project files instead of reconstructing details from conversational memory.

07

Review before approving a change.

Check the proposed destination and action before approving meaningful file operations.

08

Separate working from approved material.

Make experiments, references, current work, approved results, and archives visibly different.

09

Start a new conversation without starting over.

Review the latest continuity and continue from the recorded next step.

10

Back up the Workshop like real work.

Keep a current backup on separate storage and verify that valuable files can be restored.

Requires Flight Command

Prompts that put FCA to work

These depend on Flight Command's connection to an active project, continuity, and real Workshop files. ChatGPT alone cannot perform the FCA actions they request.

Return to an existing project

Use at the beginning of a new ChatGPT conversation.

Connect to Flight Command. Verify the active project, read the latest project continuity, and identify the current status, settled decisions, unresolved questions, and next intended step. Do not change any files yet.

Retrieve before answering

Use when the answer should already exist in the Workshop.

Search the active Flight Command project for the established source material about this question. Retrieve and use the relevant files before answering. Clearly distinguish what the project records say from any new suggestion or inference.

Continue the real work

Use after continuity is reviewed and the next step is known.

Using the active project files and latest continuity as the authority, continue the recorded next step. Preserve settled decisions and existing structure. Show me any meaningful file operation for approval before it is performed.

Save a proper continuity checkpoint

Use after a session, breakthrough, or important decision.

Create a detailed continuity checkpoint for the active Flight Command project. Record what changed, what was verified, settled decisions, files created or updated, unresolved issues, and the exact next step. Save it to the project's continuity area with a clear date and title.

Audit a project against its own records

Detect drift, contradictions, missing files, or stale instructions.

Audit the active Flight Command project using its continuity and actual Workshop files. Identify contradictions, duplicated or obsolete instructions, missing expected files, unresolved decisions, and places where current work has drifted from the latest verified direction. Do not repair anything until I review the findings.

Prepare a clean handoff

Use before changing chats, devices, collaborators, or phases.

Prepare a self-contained handoff for the active Flight Command project. Include the purpose, current state, working structure, settled decisions, exact file locations, verified results, known failures, constraints, and next actions. Save the handoff in the Workshop and report where it was placed.

Compare the actual versions

Use when choosing drafts or checking what changed.

Retrieve the relevant versions from the active Flight Command project and compare the actual files. Summarize meaningful differences, what each version preserves or loses, and which best matches the latest settled project direction. Do not rely on conversational memory.

Back up before a risky change

Use before a rebuild, migration, large edit, or experiment.

Before we make this change, use Flight Command to create a current backup of the active work and a continuity checkpoint describing the pre-change state. Verify the backup result, then stop and report the receipt before proceeding.

Branch this work

Explore a new direction without disturbing the established project.

Preserve a verified checkpoint of the active Flight Command project, then create a separate branch project for this new direction. Carry forward only the continuity, source files, constraints, and settled decisions needed for the experiment. Do not alter the original project.

Return the branch result

Bring proven work back without blindly overwriting the original.

Compare this Flight Command branch with the original project using their actual files and continuity. Identify what changed, what proved useful, what should be rejected, and any conflicts with settled decisions. Show me a proposed return plan and wait for approval before bringing selected results into the original project.

Trace this decision

Find where a project decision came from and why it was made.

Search the active Flight Command project's continuity, notes, and relevant files for this decision. Report where it first appeared, the evidence and reasoning that supported it, later references or changes, and whether it is still the current settled direction. Cite the Workshop file locations used.

Record the replacement decision

Replace an old direction without erasing its history.

Before changing this settled project decision, retrieve the existing decision and its reasoning. Record the new decision, why it replaces the old one, the evidence supporting the change, affected files, and the effective date. Preserve the former decision as superseded rather than deleting its history.

Freeze this version

Protect an approved state before experimentation or major revision.

Use Flight Command to preserve the current approved version of the relevant files and create a continuity checkpoint describing exactly what is being frozen, why it is trusted, and what experiment or revision comes next. Verify the saved version and report its location before changing anything.

Restore the last verified version

Recover the approved state after an experiment fails or drifts.

Locate the last verified approved version in the active Flight Command project. Compare it with the current file, explain what would be lost by restoring it, and show me the proposed restore operation. Do not replace the current file until I approve the exact version and destination.

Archive this project

Close completed work while keeping it available for future reference.

Prepare the active Flight Command project for archiving. Verify its final status, preserve a closing continuity record, identify the final approved deliverables, note unresolved items, and confirm the project can be understood later. Show me the archive plan and wait for approval before moving anything.

Trash this project

Remove an unwanted project through a recoverable path.

Prepare to move this Flight Command project to Trash. Verify the exact project identity, summarize what it contains, identify any files referenced by other active projects, and warn me about anything that may be lost. Require my explicit confirmation before moving the project. Do not permanently delete or empty Trash.

Recover this project from Trash

Restore a project that was removed but not permanently deleted.

Search Flight Command Trash for the project I describe. Verify its identity, original location, deletion information, contents, and any naming conflict at the restore destination. Show me the proposed recovery location and wait for approval before restoring it.

Sweep the Workshop for a missing file

Search beyond the active project when a file cannot be found.

Perform a full Flight Command Workshop search for the missing file or material I describe. Search filenames, likely alternate names, relevant folders, archived projects, continuity references, and Trash where available. Do not move or modify anything. Return the best matches with their exact locations and explain why each may be the file I need.

The bottom line

Treat Flight Command as a working shop: give each job a place, keep source material close, mark what is approved, record where you stopped, and back up the work worth keeping.

Useful before you reach for AI

Practical Math You Can Do Yourself

AI is useful, but not every problem needs AI. A few reliable shortcuts can answer everyday questions faster, keep your judgment sharp, and leave the difficult work for the tools built to handle it.

Tips and quick percentages

Move the decimal one place left to find 10%. Half of that is 5%. Double it for 20%.

$126.48 → 10% = $12.65 · 15% = $18.97 · 20% = $25.30

For 15%, add 10% and 5%. Keep the full numbers while calculating, then round the final amount. Randy's rule of thumb: round a tip up.

Sheet goods at a glance

A standard full sheet is 4 × 8 feet, or 48 × 96 inches.

Full: 4′ × 8′ · Half: 4′ × 4′ · Quarter: 2′ × 4′

That is 32, 16, or 8 square feet. Sketch the cuts before buying, account for saw-kerf and defects, and round up to whole sheets.

Lumber sizes

A label such as 2 × 4 × 8 means nominal thickness × nominal width × length in feet.

2 × 4 lumber commonly measures about 1½ × 3½ inches

Nominal and actual dimensions differ. Measure when fit matters, plan around available stock lengths, include cuts and defects, and round up to whole boards.

Square footage and materials

Area = length × width. Keep both measurements in the same unit.

12 ft × 10 ft = 120 ft²

Divide the total area by the usable coverage of one sheet, tile, box, or roll. Round up to a purchasable quantity and add an appropriate waste allowance.

Concrete for a pad

Cubic feet = length ft × width ft × depth inches ÷ 12.

Cubic yards = length ft × width ft × depth inches ÷ 324

For bags, divide required cubic feet by the yield printed on that bag. Bag yield varies by product. Round up to whole bags or the supplier's order quantity and allow for waste.

Square a layout

Use a 3-4-5 triangle: measure 3 units on one side and 4 on the other.

When the diagonal is 5 matching units, the corner is square

Scale it up when useful: 6-8-10, 9-12-15, and so on. Recheck both diagonals on a rectangular layout before committing materials.

Practical reference

Common Formula Library

Useful equations from everyday shopping and construction through mechanics, electricity, flight, and thermodynamics. Keep units consistent, state assumptions, and verify safety-critical calculations with a qualified professional.

Core constants

Formula

Pi

π ≈ 3.14159265

The constant used for circles, cylinders, spheres, rotation, and anything round.

Formula

Euler's number

e ≈ 2.71828183

The natural-growth constant used in continuous growth, decay, logarithms, and probability.

Formula

Golden ratio

φ ≈ 1.61803399

A proportion used in geometry, design, composition, and naturally occurring growth patterns.

Formula

Standard gravity

g ≈ 9.80665 m/s² = 32.174 ft/s²

The standard acceleration used for weight, falling objects, energy, and engineering estimates.

Formula

Water density

ρ ≈ 1,000 kg/m³ ≈ 8.34 lb/US gal

A practical approximation for estimating water mass, weight, storage, and pumping loads.

Everyday and business math

Formula

Percentage

percentage = part ÷ whole × 100

Find what portion one amount represents of another.

Formula

Percentage change

% change = (new − old) ÷ old × 100

Measure an increase or decrease relative to the original value.

Formula

Price after discount

final price = price × (1 − discount rate)

Calculate a sale price when the discount is written as a decimal.

Formula

Price after tax

total = price × (1 + tax rate)

Add sales tax or another percentage-based charge.

Formula

Markup price

price = cost × (1 + markup rate)

Set a selling price by adding markup to cost.

Formula

Price for a desired margin

price = cost ÷ (1 − margin rate)

Set a price that produces a target gross margin—not merely a markup.

Formula

Profit

profit = revenue − total cost

Find what remains after all included costs are subtracted.

Formula

Profit margin

margin = profit ÷ revenue × 100

Express profit as a percentage of revenue.

Formula

Break-even quantity

units = fixed costs ÷ (price − variable cost)

Estimate how many units must sell before profit begins.

Formula

Average

average = sum of values ÷ number of values

Find the arithmetic mean of a group of values.

Formula

Rate

rate = amount ÷ time

Find output, consumption, speed, pay, or another quantity per unit of time.

Geometry, construction, and materials

Formula

Rectangle area

A = length × width

Find floor, wall, sign, panel, field, or sheet area.

Formula

Rectangle perimeter

P = 2(length + width)

Find the total distance around a rectangle.

Formula

Triangle area

A = base × height ÷ 2

Find the area of a triangle using perpendicular height.

Formula

Pythagorean theorem

a² + b² = c²

Find a missing side, diagonal, or square layout in a right triangle.

Formula

Circle circumference

C = 2πr = πd

Find the distance around a circle.

Formula

Circle area

A = πr²

Find the area inside a circle.

Formula

Cylinder volume

V = πr²h

Find the capacity of a round tank, pipe, cup, or cylinder.

Formula

Rectangular volume

V = length × width × height

Find box, room, bin, or rectangular tank volume.

Formula

Sphere volume

V = 4πr³ ÷ 3

Find the volume inside a sphere.

Formula

Square feet from inches

ft² = width in × height in ÷ 144

Convert rectangular dimensions in inches into square feet.

Formula

Board feet

board ft = thickness in × width in × length ft ÷ 12

Measure lumber volume. If all three dimensions are inches, divide by 144.

Formula

Concrete cubic yards

yd³ = length ft × width ft × depth ft ÷ 27

Estimate concrete volume after expressing depth in feet.

Formula

Slope percentage

slope % = rise ÷ run × 100

Calculate grade, drainage slope, or incline.

Formula

Waste allowance

order amount = required amount × (1 + waste rate)

Add cutting, breakage, pattern, or installation waste.

Personal finance

Formula

Simple interest

I = principal × rate × time

Calculate interest when unpaid interest does not itself earn interest.

Formula

Compound amount

A = P(1 + r/n)^(nt)

Estimate savings growth or debt when interest compounds periodically.

Formula

Loan payment

M = P[r(1+r)^n] ÷ [(1+r)^n − 1]

Calculate a fixed payment using the periodic interest rate and number of payments.

Formula

Return on investment

ROI = net gain ÷ investment × 100

Compare net return with the amount invested.

Formula

Rule of 72

years to double ≈ 72 ÷ annual interest rate %

Make a quick estimate of doubling time at a steady compound rate.

Motion and mechanics

Formula

Speed

v = distance ÷ time

Find travel speed, feed rate, or movement per unit of time.

Formula

Acceleration

a = change in velocity ÷ time

Measure how quickly velocity changes.

Formula

Newton's second law

F = ma

Relate force, mass, and acceleration.

Formula

Weight

W = mg

Convert mass into gravitational force using local acceleration due to gravity.

Formula

Momentum

p = mv

Measure the motion carried by a moving mass.

Formula

Kinetic energy

KE = ½mv²

Calculate energy carried by motion; velocity has a squared effect.

Formula

Potential energy

PE = mgh

Calculate stored gravitational energy relative to a chosen height.

Formula

Work

work = force × distance

Calculate energy transferred by force acting through distance in the force direction.

Formula

Power

power = work ÷ time

Measure how quickly work is performed or energy is transferred.

Formula

Torque

τ = force × perpendicular distance

Calculate turning effect about a shaft or pivot.

Formula

Mechanical efficiency

η = useful output ÷ input × 100

Compare useful mechanical output with total input.

Fluids, aircraft, and electricity

Formula

Density

ρ = mass ÷ volume

Relate how much matter occupies a given volume.

Formula

Pressure

P = force ÷ area

Calculate force distributed across an area.

Formula

Fluid flow rate

Q = area × velocity

Estimate ideal volumetric flow through a known cross-section.

Formula

Hydraulic force

F = pressure × piston area

Estimate ideal cylinder force before losses and mechanical limits.

Formula

Dynamic pressure

q = ½ρV²

Calculate the motion-related pressure used in aerodynamic loads.

Formula

Lift

L = ½ρV²SCₗ

Relate air density, speed, wing area, and lift coefficient.

Formula

Drag

D = ½ρV²SCᴅ

Relate air density, speed, reference area, and drag coefficient.

Formula

Wing loading

wing loading = aircraft weight ÷ wing area

Compare aircraft weight with supporting wing area.

Formula

Ohm's law

V = IR

Relate voltage, current, and resistance.

Formula

Electrical power

P = VI = I²R = V²/R

Calculate electrical power when the needed values are known.

Formula

Electrical energy

E = power × time

Calculate energy use in watt-hours, kilowatt-hours, or joules with consistent units.

Formula

Series resistance

Rₜ = R₁ + R₂ + …

Add resistors connected in series.

Formula

Parallel resistance

1/Rₜ = 1/R₁ + 1/R₂ + …

Find equivalent resistance for parallel branches.

Formula

Battery runtime estimate

runtime hours ≈ watt-hours ÷ load watts

Estimate ideal runtime before conversion losses, reserve, and battery limits.

Heat and thermodynamics

Formula

Celsius to Fahrenheit

°F = °C × 9/5 + 32

Convert a Celsius temperature to Fahrenheit.

Formula

Fahrenheit to Celsius

°C = (°F − 32) × 5/9

Convert a Fahrenheit temperature to Celsius.

Formula

Celsius to Kelvin

K = °C + 273.15

Convert Celsius to the absolute temperature scale used in scientific formulas.

Formula

Sensible heat

Q = mcΔT

Calculate heat needed to change temperature without a phase change.

Formula

Latent heat

Q = mL

Calculate heat absorbed or released during a phase change.

Formula

Ideal gas law

PV = nRT

Relate pressure, volume, amount of gas, and absolute temperature for an ideal gas.

Formula

First law of thermodynamics

ΔU = Q − W

Internal-energy change equals heat added minus work performed by the system.

Formula

Second law of thermodynamics

ΔS ≥ 0 for an isolated system

Express the tendency of total entropy not to decrease in an isolated system.

Formula

Heat-engine efficiency

η = useful work output ÷ heat input

Compare useful work with supplied thermal energy.

Formula

Conduction rate

Q̇ = kAΔT ÷ L

Estimate steady one-dimensional heat flow through a flat material layer.

Formula

Convection rate

Q̇ = hAΔT

Estimate heat transfer between a surface and moving fluid.

Formula

Thermal radiation

P = εσA(T⁴ − T surroundings⁴)

Estimate net thermal radiation using absolute temperature in kelvin.

Formula

Coefficient of performance

COP = useful heating or cooling ÷ work input

Measure heating or cooling moved per unit of work supplied.

Statistics and probability

Formula

Mean

mean = Σx ÷ n

Find the arithmetic average of a dataset.

Formula

Range

range = maximum − minimum

Measure the full spread between the smallest and largest values.

Formula

Population variance

variance = Σ(x − mean)² ÷ n

Measure average squared distance from the population mean.

Formula

Population standard deviation

σ = √variance

Express population spread in the same units as the original data.

Formula

Weighted average

average = Σ(value × weight) ÷ Σweights

Average values that do not all carry equal importance.

Formula

Basic probability

probability = desired equally likely outcomes ÷ possible outcomes

Calculate simple probability when outcomes are equally likely.

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