How to use it
- Choose the Channel shape, then type the Bottom width b, m and the Side slope z, horizontal per 1 vertical for a trapezoid, or the Pipe diameter D, m for a circular pipe.
- Type the Flow depth y, m, the Bed slope S, m per m and the Roughness coefficient n.
- Read the Discharge Q, m³/s and the Mean velocity V, m/s.
- Open Working, step by step to see the flow area, the wetted perimeter and the hydraulic radius that were used.
Manning's equation
Manning's equation gives the mean velocity of water flowing steadily in an open channel, from the hydraulic radius R, the bed slope S and the roughness n. The discharge is the flow area times that velocity. In SI units the constant in front of 1/n is 1.0; the US form with 1.486 and feet gives the same result.
V = (1 ÷ n) × R^(2/3) × S^(1/2)
Q = A × V
R = A ÷ P
V mean velocity, m/s
n roughness coefficient
R hydraulic radius, m
S bed slope, m per m
Q discharge, m³/s
A flow area, m²
P wetted perimeter, mThe geometry of the three shapes
φ is the central angle of the wetted part of a circular pipe, in radians. At full depth φ is 2π and the hydraulic radius is D ÷ 4.
| Shape | Flow area A | Wetted perimeter P |
|---|---|---|
| Rectangle | b × y | b + 2 × y |
| Trapezoid | (b + z × y) × y | b + 2 × y × √(1 + z²) |
| Circular, part full | D² ÷ 8 × (φ − sin φ) | D × φ ÷ 2, with φ = 2 × arccos(1 − 2 × y ÷ D) |
Roughness values
The roughness coefficient n depends on the bed or lining. USGS Water-Supply Paper 2339 gives base values for stable channels, and the McGraw-Hill chapter on partly full pipes takes n = 0.015 for a brick channel in its worked example.
| Bed or lining | n |
|---|---|
| Concrete, stable straight channel (USGS) | 0.012 to 0.018 |
| Brick channel (textbook example) | 0.015 |
What this calculator does not include
- Natural channels with irregular sections, vegetation and flood plains. The USGS guide describes how to choose n for them, and this calculator does not do that step.
- Non-uniform flow, such as backwater behind a weir or a change of depth along the channel.
- The sizing of a channel for a design flow, with freeboard and erosion limits.
Worked example: trapezoidal channel, 2 m bottom, side slope 1.5, 1.2 m deep
With a bottom width b of 2 m, a side slope z of 1.5 and a depth y of 1.2 m, the flow area is A = (2 + 1.5 × 1.2) × 1.2 = 4.56 m². The wetted perimeter is P = 2 + 2 × 1.2 × √(1 + 1.5²) = 6.327 m, so the hydraulic radius is R = 4.56 ÷ 6.327 = 0.72076 m.
With a roughness n of 0.015 and a bed slope S of 0.001, the velocity is V = (1 ÷ 0.015) × 0.72076^(2/3) × √0.001 = 1.695 m/s, and the discharge is Q = 4.56 × 1.695 = 7.728 m³/s.
| Figure | Value |
|---|---|
| Flow area A | 4.560 m² |
| Wetted perimeter P | 6.327 m |
| Hydraulic radius R | 0.721 m |
| Mean velocity V | 1.695 m/s |
| Discharge Q | 7.728 m³/s |
Frequently asked questions
Why is there a 1.0 in the SI form of Manning's equation?
It is the unit conversion of the US form. The 1.486 of the US form, used with feet and seconds, becomes 1.0 with metres and seconds, because 1.486 × 0.3048^(1/3) is 1.000. The McGraw-Hill chapter writes the SI form as 1.0 divided by n.
What roughness n should I use for concrete?
USGS Water-Supply Paper 2339 gives 0.012 to 0.018 for a stable straight concrete channel. Choose within that range for the condition of your channel, and check the value against the design rules of the authority that governs the project.
Why must the depth in a circular pipe be less than its diameter?
Water cannot stand higher than the inside of the pipe. A depth equal to the diameter is a full pipe, and the calculator accepts that value, while a larger depth is refused.
What is the hydraulic radius?
It is the flow area divided by the wetted perimeter, R = A ÷ P. A channel with a large area for its wetted perimeter has a large hydraulic radius, and Manning's equation gives a higher velocity for it.
Why does a part-full pipe carry more water before it is full?
The friction grows with the wetted perimeter, while the flow area keeps growing as the pipe fills. The McGraw-Hill chapter on partly full pipes states that the maximum velocity and flow rate occur before the pipe is completely full.
Are my numbers sent to a server?
No. The calculation runs in your browser, and nothing you type is stored or sent to us.
Sources
- USGS Water-Supply Paper 2339, Guide for Selecting Manning's Roughness Coefficients (equation 1, Table 1)
- McGraw-Hill Higher Education, free chapter 10: Uniform Flow in a Partly Full, Circular Pipe (Example 10.3)
- IIT Madras NPTEL, Hydraulics (Prof. B. S. Thandaveswara), unit 2: channels and their geometric properties
Figures and text last checked on Thursday, November 12, 2026.