What Crouch's formula does and where it came from
Crouch's formula is the standard first-pass answer to “how fast will this boat go?” for a planing hull. It says speed is proportional to the square root of the power-to-weight ratio, scaled by a constant that characterises the hull type: V = C√(HP÷D). Rearranged, it tells you the power a target speed demands: HP = D(V÷C)².
The relationship is empirical. It was popularised by the American naval architect George Crouch and has been carried forward in design references ever since, notably Dave Gerr's The Propeller Handbook. It is not derived from resistance theory; it is a curve fit across a large number of real boats, and the constant C is the fitted parameter that absorbs everything the formula does not model — hull efficiency, running trim, appendage drag, propeller efficiency and the unit system all at once.
Two consequences dominate everything else. First, speed follows the square root of power. Doubling the horsepower produces only a 41% speed increase; going from 30 to 40 mph on the same hull requires 78% more power. That single fact explains why repowering rarely delivers what owners hope, and why the last few miles per hour of a boat's performance are so expensive. Second, speed follows the inverse square root of weight. Adding 20% to the displacement costs about 9% of the speed, which is why a boat that flies with two people aboard feels sluggish with six plus full tanks.
Choosing the constant, and what displacement really means
C is the whole model. Everything about the hull's quality is compressed into it, so choosing it correctly matters more than any other input. The commonly tabulated values run roughly: 150 for average runabouts, cruisers and passenger vessels; around 170 for light, high-speed cruisers; 190 for high-speed runabouts and sport boats; 210 for race-boat hull forms; and 220 and above for three-point hydroplanes and racing catamarans. If you know your own boat's speed at a known load and power, solve for C directly — C = V ÷ √(HP÷D) — and use that number for every later question about it. A measured C is worth far more than a table value.
Displacement means everything afloat. This is where most estimates go wrong. Use the loaded weight: bare hull, engine and drive, batteries, fuel, water, ice, gear, canvas, tender and every person aboard. A 20-foot runabout with a dry weight of 2,400 lb typically floats at 3,600 to 4,200 lb ready for a day out. Fuel alone is 6.1 lb per gallon of gasoline; a 60-gallon tank is 366 lb.
Power means shaft power. Outboard motors have been rated at the propeller shaft since the industry adopted that convention, so an outboard's rated horsepower goes in directly. Inboard engines are usually rated at the crankshaft, and a marine gearbox absorbs roughly 3%, so subtract that before entering the number. Do not use peak dynamometer figures or advertised “boosted” ratings.
The speed-length ratio is the validity check. Divide the speed in knots by the square root of the waterline length in feet. Below about 2.5 the boat is not planing, and this formula has nothing to say about it — the hull speed calculator covers that regime instead. Between 2.5 and about 3.5 the hull is in transition, climbing onto the plane, and resistance changes so rapidly with trim and load that no simple formula is reliable. Above 3.5 the boat is fully planing and Crouch's constants apply.
Worked example: a 4,000 lb runabout
Take a 20-foot runabout that floats at 4,000 lb loaded, with a conventional hull, so C = 150. You want 30 mph.
- Divide speed by the constant. 30 ÷ 150 = 0.200.
- Square it. 0.200² = 0.0400.
- Multiply by displacement. 4,000 × 0.0400 = 160 hp at the propeller shaft.
- Check it in reverse. With 160 hp installed, HP÷D = 160 ÷ 4,000 = 0.0400, √0.0400 = 0.200, and 150 × 0.200 = 30.0 mph. The two directions agree, as they must.
- Weight-to-power ratio. 4,000 ÷ 160 = 25 lb/hp, comfortably inside the range where a hull planes readily.
- Planing check. 30 mph is 30 ÷ 1.15078 = 26.07 kt. With a 20 ft waterline, √20 = 4.472, so the speed-length ratio is 26.07 ÷ 4.472 = 5.83 — fully planing.
Now ask what 40 mph would take. 40 ÷ 150 = 0.2667, squared is 0.07111, times 4,000 gives 284.4 hp. The extra 10 mph costs 124 hp, or 78% more engine, for a 33% speed gain. That is the square law in action, and it is the arithmetic every repower conversation eventually runs into.
Finally, add four passengers and a full tank — call it 900 lb, taking displacement to 4,900 lb — and keep the original 160 hp. Speed becomes 150 × √(160 ÷ 4,900) = 150 × √0.032653 = 150 × 0.180701 = 27.1 mph. The 22.5% weight increase cost 2.9 mph, or 9.6% of the speed, which matches the inverse-square-root rule: √(4,000÷4,900) = 0.9035.
How to read the result
Treat the answer as a target with a tolerance, not a specification. Crouch's formula is a fit across many boats, and a specific hull can sit several percent either side of the curve. A realistic expectation is that the predicted speed lands within about 10% of reality when C has been chosen sensibly, and within a few percent when C has been measured on the boat itself.
The weight-to-power ratio is the quickest sanity check. Under about 40 lb/hp a conventional planing hull gets onto the plane without drama. Between 40 and 70 lb/hp it depends on hull form, trim and how the load is distributed, and a boat in that band can plane empty and refuse to plane loaded. Above 70 lb/hp most planing hulls will not climb over their own bow wave at all, and the boat becomes a semi-displacement craft whether or not it was designed as one.
If the required power exceeds what the transom is rated for, stop. The maximum horsepower on the capacity plate of a monohull under 20 feet is set under 33 CFR Part 183, and exceeding it is both a safety problem and, in most US waters, an enforcement one. When the required power is out of reach, the productive lever is almost always weight, because speed responds to weight and power identically in exponent terms — both under a square root.
Speed-length ratio below 2.5 invalidates the answer, not just weakens it. If the calculator flags it, the boat is not the kind of craft this formula describes. A heavy displacement cruiser will not go faster than roughly 1.34√LWL knots no matter what engine you fit, and the extra power goes into a larger stern wave rather than speed.
Crouch constants by hull type, and what they mean at 4,000 lb
| Hull type | C | hp for 30 mph at 4,000 lb | Speed at 160 hp (mph) |
|---|---|---|---|
| Average runabouts, cruisers, passenger vessels | 150 | 160.0 | 30.0 |
| Light high-speed cruisers | 170 | 124.6 | 34.0 |
| High-speed runabouts and sport boats | 190 | 99.7 | 38.0 |
| Race-boat hull forms | 210 | 81.6 | 42.0 |
| Three-point hydroplanes, racing catamarans | 220 | 74.4 | 44.0 |
Speed at fixed power is directly proportional to C, because √(HP/D) is unchanged; power at fixed speed goes with 1/C². The constants are those tabulated in standard powering references such as Gerr's Propeller Handbook.
What this estimate does not include
- Propeller match. The formula assumes the propeller lets the engine reach its rated RPM at wide-open throttle. An overpitched wheel that holds the engine 500 RPM below its rated speed can cost 10% of the predicted top speed and shorten engine life.
- Running trim. Trim tabs and drive angle change wetted surface and therefore drag substantially. Two identical boats can differ by several miles per hour purely on trim.
- Bottom condition. Slime and light fouling add drag out of proportion to their thickness. A boat left in the water through a season commonly loses several miles per hour.
- Sea state and wind. Crouch's constants describe flat water. Chop forces you to back off, and no formula predicts the speed you will actually choose to run.
- Altitude and air temperature. Naturally aspirated engines lose roughly 3% of their power per 1,000 ft of elevation. A boat at a 5,000 ft mountain lake sees around 15% less power than its rating, which costs about 8% of top speed.
- Hull form outside the tabulated types. Stepped hulls, surface drives and air-entrapment tunnels behave differently enough that the constant must come from measurement on that specific type.
When to use a different method
Crouch's formula is the right tool for a planing monohull at the concept stage or for a repower decision. It is the wrong tool in three situations. For a displacement hull — a trawler, a heavy cruising sailboat under power, a workboat — the limit is wave-making, and the hull speed calculator gives the ceiling that no reasonable amount of power will beat. For a detailed resistance prediction on a specific planing hull, the standard method is Savitsky's, which works from deadrise angle, longitudinal centre of gravity, beam and running trim, and predicts resistance rather than fitting a constant. For a boat you already own, a measured GPS speed at a known load and RPM beats every formula.
The planning around a powering decision reaches beyond the engine itself. Once you know the top speed, the range at cruise follows from fuel burn, and the anchorages you can reach in a day follow from that; when you get there, the anchor rode and scope calculator tells you how much ground tackle to deploy for the depth and tide you find. Plotting the route or reading a waypoint off a chart plotter usually means converting between coordinate formats, which the GPS coordinate converter handles, and knowing how far ahead you can expect to see a light or a headland is a matter for the distance to horizon calculator.
One last note on units. Crouch's constants are defined for statute miles per hour, not knots, which is unusual in a marine context and is a persistent source of error. If you feed knots into the formula with a table value of C, the answer comes out 15% optimistic. This calculator converts whatever unit you choose into statute miles per hour before applying the constant, and reports the resulting speed in both mph and knots.