Cycling speed and power calculator
How many watts for a speed, how fast at a power, how long up a hill.
Still air, steady speed. CdA and Crr are examples: replace them with your own numbers.
What the numbers mean
On a bicycle your legs do work against four things: air, the road surface, gravity and the machine itself. The page adds those up for a steady speed in still air and shows how much of your power each takes. On flat ground the air dominates and the cost grows quickly with speed; on a steep hill gravity dominates, so weight matters far more than aerodynamics.
The formula
Pedal power is the sum of four forces times speed, divided by the drivetrain efficiency: P = [ ½·ρ·CdA·V³ + Crr·m·g·cosθ·V + V·(91 + 8.7·V)/1000 + m·g·sinθ·V ] / E. In words: ρ is air density in kg/m³, CdA is the drag area in m² (frontal area times drag coefficient), V is speed in m/s, Crr is the rolling resistance coefficient, m is the total mass in kg, g is 9.81 m/s², θ is the road angle (the arctangent of the grade) and E is the drivetrain efficiency, one minus your loss. The wheel bearing term and the efficiency come from the same paper: Martin and colleagues measured a chain efficiency of 97.6%, which this page shows as a 2.4% loss. The solver for speed and hill time finds the speed whose power equals yours.
How much power does it take to ride at a given speed?
It depends mostly on air drag, which grows with the cube of speed. Riding 18 mph on flat road in calm air takes about 134 W for a 190 lb rider and bike with a drag area (CdA) of 0.32 m² and a rolling resistance coefficient (Crr) of 0.004. Of that, about 76% goes to air, 20% to tire rolling resistance and the rest to the drivetrain and wheel bearings.
How fast will I go at a given power?
With the same rider and 200 W on flat road, the speed is 21.0 mph. Because drag rises with the cube of speed, a 10% increase in power (200 to 220 W) raises speed by only 3.6% on the flat. On a 6% climb most of the power fights gravity, which rises in direct proportion to speed, so the same 10% extra power gives 9.1% more speed.
What CdA and Crr should I use?
Neither can be read off a label. Martin and colleagues measured drag area in a wind tunnel for their racing-position riders and found 0.255 to 0.269 m², and used 0.0032 for the rolling resistance coefficient of their tires, quoting a published range from 0.0016 for a silk track tire to 0.0066 for a touring tire. Riders sitting upright on the hoods are less aerodynamic than riders in a racing tuck, and this page lets you enter any value. The default of 0.32 m² and 0.004 are round examples, not measurements of you.
How much does the drag area matter?
At 200 W on flat road the rider above does 22.4 mph with a CdA of 0.26 m², 21.0 mph at 0.32 and 19.6 mph at 0.40. Martin and colleagues found a similar effect in their own modeling: speed changed nearly in a straight line with drag area over the range they tried (0.211 to 0.317 m²).
Worked example: a 3-mile climb at 6 percent
For the same 190 lb rider holding 200 W up 3 miles of 6% grade, the model gives 7.7 mph, a time of 23:16 and a climb of 949 ft, or a climbing rate of about 2,446 ft per hour. Carrying 10 lb more weight at the same power makes it 24:22: 65 seconds slower.
What does altitude and temperature do?
Thinner or warmer air means less drag. The calculator takes your temperature and elevation and works out the air density from the standard atmosphere formula. At 59°F and sea level the density is 1.225 kg/m³; at 1,500 m (about 4,900 ft) and the same temperature it is 1.022, and the 200 W rider speeds up from 21.0 to 22.2 mph. The formula assumes dry air, so humidity (which makes air very slightly lighter) is ignored.
Limits of this model
It describes steady riding on a straight, smooth road in still air. Wind is not included, and Martin and colleagues showed that wind and the yaw angle of the airflow change both drag and speed; acceleration, braking, cornering and drafting are not included either. Tire pressure, road surface and wet roads change Crr well beyond the range of a single number. In their validation against a crank power meter the model matched with a standard error of 2.7 W, which is good, but their riders were on smooth roads with measured drag, rolling and air properties. With a guessed CdA, expect errors of several percent in speed.
Does this replace a power meter?
No. A model needs numbers you may not know, while a power meter measures what your legs do. The calculator is for planning, such as what power a hill needs or how much a more aerodynamic position could save, and for sanity-checking numbers from rides.
Sources and related calculators
Sources: J. C. Martin, D. L. Milliken, J. E. Cobb, K. L. McFadden and A. R. Coggan, “Validation of a mathematical model for road cycling power,” Journal of Applied Biomechanics 14(3):276–291 (1998), equations 4 to 15, the sample calculation in the appendix and the sections on drag area and rolling resistance; the density formula is the International Standard Atmosphere (troposphere layer). Wind and drafting are outside the model. Related: cycling power zones calculator, bike gear calculator, pace calculator, speed converter and tire size calculator.