What this calculator does
Hydraulic radius is a single number engineers use to describe how efficiently a channel or pipe moves water: it is the cross-sectional area of the flow divided by the wetted perimeter, the length of the boundary actually in contact with the water. It turns up in open-channel flow formulas such as Manning's equation, where a larger hydraulic radius (relatively more water, relatively less friction-causing boundary) means faster flow for the same slope and roughness.
The tricky part is the wetted perimeter, because it only counts the parts of the cross-section touching water, not the whole outline of the shape. A pipe flowing completely full has a wetted perimeter equal to its full circumference, but a pipe flowing exactly half full has a wetted perimeter of only half that circumference, and the area is also cut in half, so a hydraulic radius calculator built around the wrong perimeter gives a wrong answer even when the area is right.
The formula
Choose the channel shape, enter its dimensions, and the flow area and wetted perimeter are worked out from the standard geometry for that shape before dividing one by the other. For a rectangular channel, area is width times depth and wetted perimeter is the base plus both vertical sides. For a trapezoidal channel, the sloped sides are included using the side slope. For a circular pipe, flowing full uses the whole circle; flowing half full uses only the lower semicircle for both area and perimeter.
| Term | Meaning |
|---|---|
| Hydraulic radius | Cross-sectional flow area divided by wetted perimeter, in units of length. |
| Wetted perimeter | The length of channel boundary in direct contact with the flowing water, excluding any free surface exposed to air. |
| Side slope | For a trapezoidal channel, the horizontal distance a side wall moves for every 1 unit it rises, describing how steeply the banks are angled. |
The inputs explained
| Field | What to enter |
|---|---|
| Channel shape | Pick rectangular or trapezoidal for open channels and drains, or one of the circular options for a pipe. |
| Bottom width (rectangular / trapezoidal) (m) | The bottom width of the channel, only used for the rectangular and trapezoidal shapes. |
| Flow depth (m) | How deep the water is flowing, measured vertically from the channel bed to the water surface. |
| Side slope, horizontal per 1 vertical (trapezoidal only) | Only used for the trapezoidal shape. A slope of 1 means the bank moves out 1 unit horizontally for every 1 unit of depth (a 45 degree bank); 0 gives vertical banks, matching a rectangular channel. |
| Pipe diameter (circular only) (m) | The internal diameter of the pipe, only used for the two circular options. |
When to use it
Sizing a drainage channel
A rectangular or trapezoidal drain's hydraulic radius feeds directly into Manning's equation to estimate how much flow it can carry at a given slope and surface roughness, which is the starting point for deciding whether a proposed channel is large enough.
Checking a partially full pipe
Stormwater and sewer pipes are usually designed to run partly full rather than surcharged. Comparing the hydraulic radius flowing full against flowing half full shows how the pipe's carrying efficiency changes with depth, which is not always obvious from the diameter alone.
Comparing channel shapes for the same flow area
Two channels can have the same cross-sectional area but different wetted perimeters, and so different hydraulic radii and different flow capacities, which is a useful way to compare a rectangular design against a trapezoidal one before committing to a shape.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Hydraulic radius of a rectangular channel as flow depth increases
A fixed 2 m channel width, with the water depth increasing.
| Flow depth (m) | Hydraulic radius | Flow area | Wetted perimeter |
|---|---|---|---|
| 0.25 | 0.2000 m | 0.5000 m² | 2.500 m |
| 0.50 | 0.3333 m | 1.000 m² | 3.000 m |
| 1.00 | 0.5000 m | 2.000 m² | 4.000 m |
| 1.50 | 0.6000 m | 3.000 m² | 5.000 m |
| 2.00 | 0.6667 m | 4.000 m² | 6.000 m |
| 3.00 | 0.7500 m | 6.000 m² | 8.000 m |
Circular pipe: flowing full versus flowing half full
The same diameters compared for a pipe running completely full.
| Pipe diameter (m) | Hydraulic radius | Flow area | Wetted perimeter |
|---|---|---|---|
| 0.15 | 0.0375 m | 0.0177 m² | 0.4712 m |
| 0.30 | 0.0750 m | 0.0707 m² | 0.9425 m |
| 0.50 | 0.1250 m | 0.1963 m² | 1.571 m |
| 0.75 | 0.1875 m | 0.4418 m² | 2.356 m |
| 1.00 | 0.2500 m | 0.7854 m² | 3.142 m |
| 1.50 | 0.3750 m | 1.767 m² | 4.712 m |
Questions
Is hydraulic radius the same as the actual radius of a pipe?
No, and for a pipe flowing full it is only a quarter of the pipe's actual radius: hydraulic radius equals diameter divided by 4 for a full circular pipe, not the radius itself. The name is a historical convention, not a literal measurement.
Why does a half-full pipe have the same hydraulic radius as a full one?
Because both the flow area and the wetted perimeter of the lower semicircle are exactly half of the full circle's values, so the ratio between them, which is what hydraulic radius measures, stays the same. This does not hold at other flow depths.
Why not just use the cross-sectional area on its own?
Area alone ignores friction. A wide, shallow channel and a narrow, deep one can have the same area but very different amounts of wetted boundary generating drag on the flow, and hydraulic radius is what captures that difference for use in flow-capacity formulas.
What side slope should I use for a trapezoidal channel?
It depends entirely on the design and the soil or lining material; there is no universal value. Enter whatever horizontal-to-vertical ratio the channel's banks are actually built or designed to, which is usually specified in the project drawings.
For the pipe-friction side of open-channel and pipe-flow work, see the Darcy friction factor calculator. To convert a known flow area, velocity and density into a flow rate, use the flow rate calculator.