Horsepower calculator

Where 5,252 comes from

Torque is a twisting force: a force applied at a radius, and on its own it says nothing about time. Power is work per unit time. One revolution of the crank moves that force through 2π radians, so the work per revolution is 2π × torque, and the work per minute is 2π × torque × rpm. That is already power, in foot-pounds-force per minute.

James Watt wanted a unit his customers could picture, so he defined one horsepower as 33,000 foot-pounds-force per minute — a horse lifting 330 lb by 100 ft in a minute. Dividing one by the other leaves a single constant:

hp = torque_lb·ft × rpm × 2π ÷ 33,000 = torque_lb·ft × rpm ÷ (33,000 ÷ 2π) = torque_lb·ft × rpm ÷ 5,252.113

This is also the whole story of the famous crossing point. At 5,252 rpm the multiplier rpm ÷ 5,252 equals one, so the two curves print the same number and cross. Below that speed the torque line is higher, above it the power line is. Nothing whatsoever happens inside the engine at that rpm; the crossing is a property of the unit, and it exists only because horsepower is defined in feet and pounds. Plot the same pull in kilowatts and newton-metres and the curves cross at 9,549 rpm instead, where most engines never go.

The units themselves are three different sizes of the same idea. One mechanical horsepower is 745.70 W. One PS — Pferdestärke, the metric horsepower a DIN 70020 rating is quoted in — is 735.50 W, defined as 75 kilogram-force metres per second. The kilowatt is the SI unit and needs no definition beyond itself. So 100 hp is 74.57 kW and 101.39 PS, and a European brochure quoting PS is printing a number about 1.4 % larger than the same engine's hp figure for free.

Why a formula cannot give a number

The converter above is exact because it is a definition. An estimate of power from displacement is not, and the reason is that at least five things stand between swept volume and the torque a cylinder actually makes.

• Volumetric efficiency. An engine only fills its cylinders completely at the rpm its intake and exhaust are tuned for; everywhere else it traps less air than its displacement, and above about 90 % the ports, valve area and runner length decide the number, not the bore.
• Compression ratio. Thermal efficiency rises with it, so the same trapped air makes more work at 12:1 than at 9:1 — until the end gas detonates and the calibration has to be pulled back.
• Cam timing. Late intake valve closing trades low-rpm cylinder filling for high-rpm filling. Two engines identical in every other dimension can peak 2,000 rpm apart and 80 hp apart on the cam alone.
• Boost. A compressor breaks the link between displacement and trapped mass entirely. Air mass per cycle scales roughly with absolute manifold pressure, so 1 bar of boost is worth most of a second engine.
• Friction. Rubbing and pumping losses grow with mean piston speed, which is why a long-stroke engine runs out of usable rpm before its head does, and why the last thousand rpm of any engine costs more than the first.

None of those five is in "displacement × cylinders". Every horsepower estimator on the web is therefore a lookup table with a text box in front of it: someone picked a hp-per-litre figure per category and the page multiplies. The second panel here does the same thing. The difference is that its figures are measurements, with the engines that produced them named.

What the roster measures

Every four-stroke crank engine in the simulator was run on its dyno in the same fixed air, and peak power divided by displacement gives the specific output below. The classes are the ones the roster can actually populate; the band's edges are named engines, not percentiles of an imaginary population.

Read those as what this simulator measures, not as industry norms. They came from its dyno at one fixed set of ambient conditions, with no correction applied afterwards. Presets whose description says "Same … as" a real engine reproduce that engine's published output within 5 %, so those rows are anchored to something outside the simulator; the rest are plausible engines rather than copies of a specific one. Two calibration debts are outstanding and both land on the boosted rows: exhaust temperature currently follows the throttle lever instead of combustion, and diesel and petrol fuel consumption are ordered the wrong way round. When those are fixed the turbo and supercharged bands will move, and this table moves with them because it is generated, not typed.

Two-stroke engines are excluded from both tables. A two-stroke fires every revolution instead of every second one, so its hp per litre is not comparable with anything above, and including it would widen every band for no reason.

The double-correction trap

A manufacturer's rated figure is not a raw reading. It was corrected to a reference day under SAE J1349 before it was printed, which is the whole point of the standard: it makes two engines rated in different weather comparable. So a rated figure is already a standard-day number, and multiplying it again by today's correction factor gives a number that describes no day at all.

The same applies to everything measured here. The simulator's dyno runs in fixed air, and each preset is calibrated so its peak matches a published rating — an SAE-corrected rating. Correct it a second time and the error compounds. If you want to see how far apart four correct answers to the same pull can be, the dyno correction calculator prints SAE, STD, DIN and EC side by side, and a single reading spreads about 4 % across them.

Wheel versus crank

An engine dyno measures at the flywheel. A chassis dyno measures at the tyre, after the clutch, the gearbox, the differential, the driveshafts and the tyre carcass have each taken a cut. The cut is real and it is not a constant. It depends on the gear the pull was made in, on whether the car is front, rear or all-wheel drive, on oil temperature, and on how hard the strap is pulling the car onto the rollers.

This is why "wheel hp × 1.15 = crank hp" is the most confidently wrong number in the hobby. A manual rear-drive car in fourth might lose 11 %; an all-wheel-drive car with a torque-converter automatic can lose over 25 %, and the same car loses a different share in a different gear. Compare wheel figures with wheel figures, on the same dyno in the same gear, and treat a crank figure as a separate measurement rather than a conversion. In the simulator the loss is not assumed either: the torque converter and clutch are simulated between the crank and the wheels, so the two numbers differ because something between them is doing work.

Worked example: the 7.0-litre pushrod V8

The roster's V8 7.0L Pushrod is built to a real engine's dimensions: Same bore, stroke, compression ratio and displacement as the Chevrolet LS7 7.0 V8. Run on the simulator's dyno it peaks at 502 hp at 5,688 rpm. Nothing measured that figure directly — the dyno measured torque at every rpm, and this is the point where torque × rpm was largest.

Going backwards through the identity gives the torque at the power peak: 502 × 5,252.113 ÷ 5,688 = 463.8 lb·ft. That is not the engine's peak torque, which is 648 N·m and arrives lower down at 4,638 rpm. Torque falls between the two points and rpm rises faster, which is what moves the power peak above the torque peak on every engine here.

Put it through the estimator instead and the answer is a band rather than a point: 7.01 L, petrol, naturally aspirated, road. The measured 71.6 hp per litre it actually makes sits inside that band, which is all an estimate from displacement can ever tell you. The dyno is what tells you the rest.

Questions people ask

Sources

• Britannica, "horsepower" Watt's 33,000 foot-pounds per minute, and how he arrived at it from a mill horse.
• NIST Special Publication 811, Appendix B: horsepower (mechanical) = 745.69987 W, horsepower (metric) = 735.49875 W. The two conversions this page uses.
• SAE J1349, Net Power Rating. What a manufacturer's quoted figure already has applied to it.
• DIN 70020, the German standard whose PS figures fill European brochures, and the reason a rating in PS is not a rating in hp.
• The simulator's own roster and its generated peak figures, which are where every hp-per-litre number on this page comes from.

Related reading

• Dyno correction calculator Why the same pull prints four different legitimate numbers.
• How a torque converter works The part of the drivetrain that makes wheel horsepower disagree with crank horsepower.
• The Otto cycle explained Where torque comes from in the first place, and why trapped air mass sets it.
• Every engine in the simulator Filterable by layout, fuel and induction, with the measured peaks in this page's tables.

Other calculators

• Compression ratio calculator Static, dynamic and boosted CR from bore, stroke, chamber, gasket and piston.
• Engine displacement calculator Bore × stroke × cylinders to cc, litres and cubic inches, with overbore.
• Bore and stroke calculator Solve bore or stroke for a target size, the ratio, and piston speed at redline.
• Dyno correction calculator SAE, STD, DIN and EC corrected power from one reading.

Want a hp-per-litre band for a class the roster does not cover yet? Tell us.