Understand the numbers before using the calculator
An E6B connects three ideas: motion is a vector, a rate must be multiplied by the correct time unit, and a useful result needs clearly stated assumptions. Start with a simple example you can estimate mentally, then change one input at a time. These modules are mathematical learning aids for ground-school or simulator practice. They do not establish aircraft performance, legal fuel reserves, safe runway limits or an operational flight plan.
Choose a module above. Results update when its inputs change; the Calculate button repeats the calculation. A blank required field is an error, not zero. If a result is rejected, read the explanation and correct the relevant input. The selected time-speed-distance result is calculated from the other fields. Numbers are rounded for display, while calculations retain more precision internally. General numeric inputs accept zero or magnitudes from 10−12 to 109; physical and model-specific limits are stricter where stated. Very small nonzero outputs use scientific notation rather than a misleading zero.
| Quantity | Meaning here | Common mistake |
|---|---|---|
| Course / ground track | Direction of the desired / actual path over Earth | Assuming it always equals heading |
| Heading | Direction the aircraft's nose points | Mixing a magnetic heading with a true course |
| TAS | Speed through the surrounding air mass | Using it as ground speed in a headwind |
| GS | Speed over the ground along the track | Using air distance for a ground-route problem |
| Wind from | Direction the wind comes from | Drawing its motion toward that same direction |
| kt and NM | One knot is one nautical mile per hour; 1 NM = 1,852 m | Mixing statute miles and nautical miles |
Wind triangles: the air moves too
The ground-velocity vector equals the aircraft's air-velocity vector plus the wind-velocity vector. A wind from 090° blows toward 270°. To follow a northerly course in that wind, the aircraft points slightly east of north so its eastward air-velocity component cancels the westward wind.
Use true north for the Wind / Heading and Find Wind modules. In the runway module, wind and runway direction must use the same reference. FAA guidance describes METAR wind directions as true and tower-issued surface winds as magnetic. A runway's number is rounded; it is not a substitute for the direction required by an exact exercise. Convert references before combining them.
Crosswind = wind speed × sin(Δ)
Headwind = wind speed × cos(Δ)
WCA = asin(crosswind ÷ TAS)
True heading = true course + WCA
Ground speed = TAS × cos(WCA) − headwind
Angles in these equations are in degrees. A positive crosswind component is from the right; a positive WCA is a clockwise correction. Negative headwind means tailwind. The solution uses the forward-pointing branch of the wind triangle. If the crosswind exceeds TAS, it cannot be canceled. Even if it can be canceled, the remaining forward speed may be zero or negative, so the desired forward track is unavailable. This is a mathematical feasibility result, not an aircraft operating limit.
Worked example 1: a pure crosswind still reduces ground speed
Set course 000°, TAS 120 kt, wind from 090° at 20 kt, variation 0° and deviation 0°. The crosswind is 20 kt and the headwind component is zero. WCA = asin(20 ÷ 120) = 9.594°. True heading is therefore 009.594°, or about 010° to the nearest degree. The forward component is √(120² − 20²) = 118.322 kt. It is less than 120 kt because some airspeed is being used to oppose the crosswind. The equal-scale live diagram adds the vectors head to tail.
Worked example 2: distinguish direction from speed
Heading 000° at TAS 120 kt and track 000° at GS 100 kt imply a 20 kt wind blowing south, so its wind-from direction is 000°. If both speeds are 120 kt on the same heading and track, the calculated wind is calm and its direction is undefined. Assigning an arbitrary compass direction to a zero-length vector would be misleading.
For the heading conversion convention used here, east variation and east deviation are positive: magnetic heading = true heading − variation; compass heading = magnetic heading − deviation. For example, true 010° with variation +3° and deviation −2° gives magnetic 007° and compass 009°. A deviation correction belongs to the particular compass and heading; this tool does not supply a deviation card.
Runway components: keep signs and gust assumptions clear
A wind 30° to the right of the runway at 20 kt has crosswind 20 sin 30° = 10 kt from the right and headwind 20 cos 30° ≈ 17.32 kt. If the gust speed is 30 kt from that same direction, the gust components are 15 kt crosswind and about 25.98 kt headwind. The optional gust calculation assumes an unchanged direction. It does not model variable wind, turbulence, braking action or a maximum demonstrated crosswind as a universal safe limit.
Time, speed and distance: keep the hours
Time (h) = distance (NM) ÷ GS (kt)
GS (kt) = distance (NM) ÷ time (h)
Time (h) = hours + minutes ÷ 60
Worked example 3: 145 NM at 115 kt
Time = 145 ÷ 115 = 1.260869… h. Multiplying the fractional hour by 60 gives 15.652… min. The result is about 1 h 16 min, rounded to the nearest minute, or 75.652… minutes before rounding. Do not read 1.26 h as 1 h 26 min. Conversely, 1 h 20 min is 1 + 20/60 = 1.333333… h; covering 145 NM in that time requires 108.75 kt.
When calculating time, speed must be positive. When calculating speed, elapsed time must be positive. A zero distance can legitimately give zero elapsed time or zero speed in the corresponding idealized exercise. A displayed duration of 2 h 0 min is preferable to 1 h 60 min after rounding. Check that units cancel: NM ÷ (NM/h) = h.
Fuel arithmetic: reserve is not fuel already burned
This module uses US gallons throughout, a constant supplied ground speed and a constant burn rate. It includes the entered taxi/start fuel and user-selected reserve duration. It does not add separate climb, alternate, contingency, holding or unusable-fuel allowances. “Usable fuel” is already the usable amount, not total tank capacity. Reserve time is an input for the exercise; the calculator does not decide which regulation applies.
Reserve fuel = burn rate × reserve minutes ÷ 60
Required total = taxi fuel + trip fuel + reserve fuel
Fuel at destination = usable fuel − taxi fuel − trip fuel
Surplus above reserve = fuel at destination − reserve fuel
Worked example 4: two different remaining-fuel numbers
Use 180 NM, GS 120 kt, burn 8 US gal/h, usable fuel 40 US gal, taxi fuel 1 US gal and reserve 45 min. Trip time is 1.5 h, trip fuel is 12 US gal and reserve fuel is 6 US gal. The budget requires 1 + 12 + 6 = 19 US gal. Fuel at the destination is 27 US gal; the surplus above the selected reserve is 21 US gal. Those quantities should never share an ambiguous “fuel remaining” label.
Endurance after taxi is (40 − 1) ÷ 8 = 4.875 h. Keeping 6 US gal untouched leaves 4.125 h for travel at the assumed rate, giving 4.125 × 120 = 495 NM at the supplied ground speed. This is not a no-wind aerodynamic range prediction. A fuel deficit must remain visible; clamping a negative budget to zero does not make the example feasible. For more on range assumptions, see the aircraft range calculator and worked guide.
Altitude and airspeed estimates have model limits
The pressure-altitude shortcut uses altitude in feet and altimeter setting in inches of mercury. It is a training approximation, not an exact atmospheric reduction. The density-altitude shortcut adjusts this pressure altitude for the temperature difference from a simplified standard atmosphere:
Standard temperature (°C) ≈ 15 − 1.98 × PA/1,000
DA ≈ PA + 120 × (OAT − standard temperature)
Worked example 5: a hot day at an elevated airfield
At elevation 5,000 ft, setting 29.42 inHg and OAT 30°C, the shortcut gives PA ≈ 5,500 ft. Standard temperature at that pressure altitude is 15 − 1.98 × 5.5 = 4.11°C. The temperature difference is 25.89°C, so DA ≈ 5,500 + 120 × 25.89 = 8,606.8 ft. This explains why high elevation on a hot day can behave like a still higher standard-atmosphere altitude. It does not calculate a takeoff distance or climb capability.
The separate TAS estimate uses a dry-air pressure/density model and TAS ≈ EAS/√σ, where σ is density relative to standard sea level. The input is a calibrated-airspeed approximation treated as equivalent airspeed at low speed. Raw indicated airspeed needs instrument/position correction; compressibility corrections are also omitted. Humidity is omitted. The model accepts pressure altitude from −2,000 to 36,089 ft, OAT −80 to 60°C and calibrated airspeeds up to 250 kt as computational limits, not as a guarantee of accuracy throughout that range. Use the relevant aircraft's approved data for operational work.
Climb, descent and off-course geometry
Gradient (%) = gradient (ft/NM) ÷ 6,076.11549… × 100
Path angle = atan[gradient (ft/NM) ÷ 6,076.11549…]
Worked example 6: vertical rate is not an angle
At 120 kt, ground speed is 2 NM/min. A vertical-rate magnitude of 600 ft/min produces 300 ft/NM, a gradient of about 4.94% and an angle of about 2.83°. A 3,000 ft altitude change takes 10 NM under those fixed conditions. For an exact 3° path at the same ground speed, the rate is tan 3° × 6,076.11549… × 2 ≈ 636.87 ft/min. Changing ground speed changes the rate required to preserve an angle.
The tool treats vertical speed and altitude change as nonnegative magnitudes. It does not model acceleration, turns, level-offs, terrain clearance or a procedure's constraints. Zero vertical speed cannot achieve a positive altitude change in a finite distance. An angle near 90° has an extremely large tangent and is outside the everyday use of this simplified geometry.
For off-course practice, the distance flown and remaining fields mean distances measured along the intended course. The cross-track error is perpendicular to that course. With 3 NM error after 60 NM along course and 30 NM along course remaining, drift angle = atan(3/60) ≈ 2.862°, closing angle = atan(3/30) ≈ 5.711°, and their sum is about 8.573°. Turn right in the exercise if you are left of course, or left if right. This idealized correction assumes unchanged drift; it is not a navigation instruction for changing wind. The familiar 1-in-60 estimates, 3° + 6° = 9°, are small-angle approximations. Do not use the same atan denominator for a distance measured along the slanted actual track.
Eight practice checks with explained answers
Predict the sign, units and rough size first. Reveal the solution, then reproduce it with the appropriate module. These are original learning questions, not FAA examination items or a proficiency score.
1. Course 090°, TAS 100 kt, wind from 090° at 20 kt: heading and GS?
The wind is directly ahead. WCA is 0°, true heading is 090° and GS = 100 − 20 = 80 kt. With wind from 270° instead, GS would be 120 kt.
2. Course 000°, TAS 60 kt, wind from 000° at 70 kt: can the forward track be held?
No. Crosswind is zero, but the greatest forward ground speed along that course is 60 − 70 = −10 kt. A numerical heading must not be presented as a successful forward-track solution.
3. Wind 45° from the right at 20 kt: runway components?
Headwind and right crosswind both equal 20/√2 ≈ 14.142 kt. Their squares sum to 20², a useful vector check.
4. 90 NM at 120 kt: time and fuel at 8 US gal/h?
Time = 90/120 = 0.75 h = 45 min. Trip fuel = 8 × 0.75 = 6 US gal, before any separate taxi fuel or reserve allocation.
5. 25 US gal usable, taxi 1, trip 12 and reserve 6: what remains?
Fuel at destination = 25 − 1 − 12 = 12 US gal. Surplus above reserve = 12 − 6 = 6 US gal. Total required for that budget is 19 US gal.
6. Sea-level elevation, 29.92 inHg and 15°C: the altitude shortcuts?
PA = 0 ft, standard temperature = 15°C and DA ≈ 0 ft. The dry standard-atmosphere density ratio is approximately 1, so the low-speed estimated TAS is approximately the entered EAS/CAS value.
7. GS 90 kt and vertical rate 450 ft/min: gradient and distance for 2,000 ft?
90 kt = 1.5 NM/min. Gradient = 450/1.5 = 300 ft/NM. Distance = 2,000/300 ≈ 6.667 NM under the stated constant conditions.
8. Convert 10 NM, 10 US gallons and 20°C
10 NM = 18.52 km. 10 US gal = 37.85411784 L. 20°C = 68°F = 293.15 K. Imperial gallons have a different size: 1 imperial gal = 4.54609 L.
Common questions
Can this page be used for real flight navigation?
No. Treat it as an educational calculator. Its simplified inputs omit many aircraft, weather, regulatory and route constraints. Use approved aircraft documentation, current official information and appropriately qualified instruction for operational decisions.
Why is “wind from” different from the cyan arrow?
The reported direction names the source of the wind. A velocity arrow shows where the air moves, 180° opposite. The diagram's wind arrow begins at the air-vector tip and ends at the ground-vector tip.
Why does the calculator reject my input?
Required blanks, nonfinite numbers, negative magnitudes, division by zero and values beyond the stated model limits are not meaningful inputs. Wind results are also rejected when the requested forward track is mathematically impossible. Correct the field rather than treating an error as zero.
Is the reserve a legal minimum or a safety approval?
No. It is the duration you selected for a mathematical exercise. Applicable requirements and additional allowances depend on the actual operation; the tool does not determine compliance or whether a flight is safe.
Does this page have an official FAA E6B score?
No. It is independent study material. The FAA Private Pilot Airplane knowledge-test row in the September 25, 2026 matrix lists 60 questions, 2.0 hours and a 70% passing score. Additional unscored validation questions may be included. Those numbers describe the named test, not this page's practice questions. Check current FAA and provider instructions before booking or choosing test aids.
For supporting mathematics, use the scientific calculator for degree-mode trigonometry and the scientific-notation guide for interpreting small and large quantities.
Sources & References
Primary references checked October 4, 2026. The explanations, examples and practice above are original. These references explain underlying concepts and official testing context; none endorses this calculator.
- FAA Pilot's Handbook of Aeronautical Knowledge: navigation, flight instruments and aircraft-performance chapters
- FAA AIM Chapter 7, Section 1: aviation weather reporting, including METAR conventions
- FAA AIM Chapter 4, Section 3: airport operations and tower wind information
- FAA Airman Knowledge Testing Matrix: current test lengths and passing-score context
- FAA Airman Testing: official test information and scheduling routes
- FAA Airman Certification Standards: current standards and companion guidance
- PSI FAA testing sign-in: preserved original provider link; appointment/account access is required
- FAA-hosted Aerodynamics for Naval Aviators: equivalent/true airspeed and density relationship
- NIST SI Guide, conversion factors: unit definitions and conversion conventions