Hydraulic Radius Calculator
Calculate hydraulic radius, flow area, wetted perimeter, hydraulic diameter, and key channel geometry for open-channel and pipe sections.
Calculator is for informational purposes only. Terms and Conditions
Choose the geometry
Select the channel or pipe shape before entering the required known values.
Enter the known values
Only the inputs required for the selected geometry are shown.
Visual Check
The highlighted boundary is the wetted perimeter. The open water surface is not included.
Solution
Live hydraulic radius, geometry checks, warnings, and full solution steps.
Quick checks
- Check—
Show solution steps See the geometry equations, substitutions, assumptions, and result path
- Enter values to see the full calculation steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
Uses standard hydraulic radius geometry formulas for educational and preliminary engineering calculations.
- Assumptions will appear after a valid calculation.
Calculator guide
How the Hydraulic Radius Calculator Works
The Hydraulic Radius Calculator above calculates hydraulic radius \(R_h\) from either known flow area and wetted perimeter or from the geometry of rectangular, trapezoidal, triangular, full circular, and partially full circular flow sections.
Hydraulic radius is defined as cross-sectional flow area divided by wetted perimeter. It is a geometry property used extensively in open-channel hydraulics and in resistance equations such as Manning’s equation. The key input is the wetted perimeter: only the solid boundary touching water is included. The free water surface of an open channel is excluded.
- Core formula
- \(R_h=A/P\)
- Geometry modes
- Custom area/perimeter, rectangular, trapezoidal, triangular, full pipe, and partially full pipe
- Most important rule
- Do not include the free water surface in wetted perimeter
How to Calculate Hydraulic Radius
Use the geometry mode that most directly matches the information you already have, then verify the calculated flow area and wetted perimeter before relying on \(R_h\).
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Choose the cross-section geometry
Select custom area and wetted perimeter, rectangular channel, trapezoidal channel, triangular channel, full circular pipe, or partially full circular pipe.
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Enter the dimensions that describe the water section
Use bottom width, vertical flow depth, horizontal-to-vertical side slope, inside pipe diameter, or known area and wetted perimeter as required by the selected mode.
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Check what the water actually touches
For an open channel, trace the bed and side boundaries under water. Do not trace across the free surface. For a partially full pipe, use only the wetted circular arc.
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Review the live visual and quick checks
The calculator’s visual highlights the wetted boundary, and its result area provides supporting geometry such as flow area, wetted perimeter, hydraulic diameter, top width, hydraulic depth, percent full, or central angle when applicable.
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Use the result in the next hydraulic calculation
Hydraulic radius is often carried into Manning’s equation or another resistance calculation together with area, roughness, and slope.
Calculator Inputs and Outputs
The required fields change with geometry. The live calculator also supports unit presets, individually selectable units, answer-unit selection, precision control, visual geometry checks, and solution steps.
| Quantity | Meaning | Typical Use |
|---|---|---|
| Flow area \(A\) | Cross-sectional area occupied by flowing water | Custom geometry or supporting result |
| Wetted perimeter \(P\) | Solid boundary length in contact with water | Custom geometry or supporting result |
| Bottom width \(b\) | Flat channel-bottom width | Rectangular and trapezoidal channels |
| Flow depth \(y\) | Vertical water depth from channel bottom or pipe invert | Open channels and partially full pipes |
| Side slope \(z\) | Horizontal distance per one unit vertical | Trapezoidal and triangular channels |
| Pipe diameter \(D\) | Inside diameter of the circular conduit | Full and partially full pipes |
| Hydraulic radius \(R_h\) | Flow area divided by wetted perimeter | Primary result |
| Hydraulic diameter \(D_h\) | \(4A/P=4R_h\) | Equivalent noncircular-flow dimension |
| Hydraulic depth \(D_{hyd}\) | Flow area divided by free-surface top width | Open-channel Froude-number and wave calculations |
Hydraulic Radius Formulas by Shape
Every shape follows the same relationship \(R_h=A/P\). The geometry determines how \(A\) and \(P\) are calculated.
General hydraulic radius
Area divided by wetted perimeter produces a length.
Rectangular channel
Trapezoidal channel
Here \(z\) is horizontal-to-vertical side slope, such as \(z=3\) for 3H:1V.
Triangular channel
A symmetric trapezoidal channel with \(b=0\) becomes a triangular V-channel.
Full circular pipe
Hydraulic diameter
Hydraulic diameter is four times hydraulic radius—not twice the hydraulic radius.
Worked Example: Trapezoidal Channel
Find the hydraulic radius for a trapezoidal channel with a 4 ft bottom width, 2 ft water depth, and 3H:1V side slopes on both sides.
Flow area
Wetted perimeter
Hydraulic radius
Result
Hydraulic radius \(\approx1.20\ \mathrm{ft}\)
The result is less than the 2 ft flow depth, which is reasonable for this channel geometry.
What Counts as Wetted Perimeter?
Wetted perimeter is the length of the containing solid boundary in contact with the water. USACE explicitly defines hydraulic radius as flow area divided by wetted perimeter and notes that the wetted perimeter does not include the free surface.
| Boundary | Include in \(P\)? | Why |
|---|---|---|
| Channel bottom under water | Yes | Water is in contact with a solid boundary |
| Submerged channel side | Yes | Water is in contact with the channel wall or bank |
| Open water surface | No | This is the water-air interface, not the containing boundary |
| Wetted arc of a partial pipe | Yes | The pipe wall is in contact with water |
| Dry upper arc of a partial pipe | No | No water contacts that part of the pipe wall |
Partially Full Circular Pipes
A partially full circular pipe is an open-channel geometry problem. The water depth determines a circular segment area and a wetted arc rather than the full pipe area and circumference.
Central angle
The angle \(\theta\) is in radians and corresponds to the wetted circular segment.
Area, wetted perimeter, and hydraulic radius
| Water depth | Central angle | Hydraulic-radius check |
|---|---|---|
| \(y=D/2\) | \(\theta=\pi\) | \(R_h=D/4\) |
| \(y=D\) | \(\theta=2\pi\) | \(R_h=D/4\) |
The half-full result surprises many users: both the wetted area and wetted perimeter are exactly half of their full-pipe values, so their ratio remains \(D/4\). Between half full and full, however, hydraulic radius varies nonlinearly with depth.
How to Interpret Hydraulic Radius
Hydraulic radius measures flow area relative to boundary contact. For otherwise comparable conditions, a larger \(R_h\) generally corresponds to less boundary contact per unit of flow area and therefore lower relative boundary resistance.
Wide rectangular channel
When \(b\gg y\), \(P=b+2y\approx b\), so \(R_h=by/(b+2y)\) approaches the flow depth \(y\).
Full circular pipe
The exact geometry check is \(R_h=D/4\), so hydraulic radius equals half the geometric pipe radius.
Small shallow sections
Shallow flow commonly has a relatively large wetted perimeter compared with area, producing a smaller hydraulic radius.
Hydraulic Radius vs Related Terms
Hydraulic radius, hydraulic diameter, hydraulic depth, and pipe radius are different quantities. Mixing them can produce a calculation that looks dimensionally plausible but is physically wrong.
| Term | Formula | Typical Use |
|---|---|---|
| Hydraulic radius | \(R_h=A/P\) | Open-channel and conduit resistance geometry |
| Hydraulic diameter | \(D_h=4A/P=4R_h\) | Equivalent flow dimension for noncircular conduits |
| Hydraulic depth | \(D_{hyd}=A/T\) | Free-surface calculations, especially Froude number |
| Pipe radius | \(r=D/2\) | Geometric radius of a circular pipe |
How Hydraulic Radius Is Used in Manning’s Equation
FHWA presents hydraulic radius as a cross-section property used in Manning’s equation along with area, roughness, and slope. Hydraulic radius alone is therefore an intermediate geometry result rather than the final discharge answer.
Manning discharge form
\(Q\) is discharge, \(A\) is flow area, \(R_h\) is hydraulic radius, \(S\) is energy/friction slope under the applicable uniform-flow assumption, \(n\) is Manning roughness, and \(K_u\) depends on the unit system.
The \(R_h^{2/3}\) term means hydraulic radius materially affects the Manning conveyance, but it must be evaluated together with area, roughness, and slope. Irregular sections may also need subdivision when different portions of the cross section have different roughness characteristics.
Units and Input Checks
Hydraulic radius has length units. The most damaging errors come from mixing length and area units or entering a different geometric definition than the selected mode expects.
Use inside pipe diameter
Pipe-flow geometry is based on the internal flow boundary, not nominal pipe size or outside diameter unless those happen to equal the actual inside diameter.
Enter side slope as H:V
A 3H:1V side slope means \(z=3\). Do not enter 3 degrees.
Area conversions are squared
\(1\ \mathrm{ft}=0.3048\ \mathrm{m}\), but \(1\ \mathrm{ft^2}=0.09290304\ \mathrm{m^2}\).
Partial-pipe depth must satisfy \(0<y\le D\)
Depth greater than the inside diameter cannot describe a partially full circular section.
Custom mode requires actual flow area
Do not enter total excavation area or full conduit area unless the water actually occupies that full area.
Check the visual boundary
The calculator’s live graphic is useful for confirming which channel or pipe surfaces are included in the wetted perimeter.
Common Hydraulic Radius Mistakes
The \(A/P\) equation is straightforward. Most wrong answers come from defining \(A\) or \(P\) incorrectly.
Including the free water surface
This makes wetted perimeter too large and hydraulic radius too small.
Using full circumference for a partial pipe
Only the submerged arc contributes to wetted perimeter when the pipe is not full.
Confusing hydraulic radius with pipe radius
For a full circular pipe, \(R_h=D/4=r/2\), not \(r\).
Using top width as wetted perimeter
Top width is the free-surface width. It is used in hydraulic-depth calculations, not as the wetted perimeter.
Using a simple shape for an irregular channel
Benches, compound sections, natural banks, and multiple roughness zones may require surveyed geometry and more detailed subdivision.
Assuming larger \(R_h\) automatically means adequate design
Capacity and stability still depend on area, roughness, slope, discharge, material, freeboard, and boundary conditions.
Assumptions and Engineering Limits
The calculator is a geometry tool. It does not replace a complete open-channel, stormwater, culvert, sewer, or hydraulic-design analysis.
Idealized cross-section
Each shape mode assumes the selected ideal geometry accurately represents the water section.
No roughness calculation
The calculator does not determine Manning’s \(n\) from channel material, vegetation, irregularity, or maintenance condition.
No normal-depth solution
It does not solve the coupled Manning equation for water depth from a known discharge, slope, and roughness.
No backwater or rapidly varied flow
Tailwater, controls, culvert inlets/outlets, hydraulic jumps, and water-surface profiles require additional hydraulic analysis.
No stability or erosion check
Boundary shear, permissible velocity, lining stability, sediment transport, scour, and erosion are outside this geometry calculation.
No code or permit determination
Final design should follow the governing agency criteria, project standards, field conditions, and qualified engineering review.
Sources and Calculation Basis
The definition, wetted-perimeter convention, trapezoidal geometry, and Manning-equation context were checked against current federal hydraulic references.
- FHWA — Hydraulic Design Series No. 4: Introduction to Highway Hydraulics — defines hydraulic radius as cross-sectional flow area divided by wetted perimeter and provides rectangular/trapezoidal geometry used with Manning’s equation.
- USACE Hydrologic Engineering Center — Glossary — defines hydraulic radius as flow area divided by wetted perimeter and explicitly states that wetted perimeter does not include the free surface.
The trapezoidal worked example was independently recomputed: \(A=20.000\ \mathrm{ft^2}\), \(P=16.6491\ \mathrm{ft}\), and \(R_h=1.20127\ \mathrm{ft}\). The full-pipe check \(R_h=D/4\) and half-full-pipe check \(R_h=D/4\) were also verified algebraically.
Hydraulic Radius Calculator FAQ
These answers address the distinctions that most often cause incorrect hydraulic-radius calculations.
What is the hydraulic radius formula?
Hydraulic radius is \(R_h=A/P\), where \(A\) is the cross-sectional flow area and \(P\) is the wetted perimeter.
Does wetted perimeter include the water surface?
No. For an open channel, the free water surface is excluded. Wetted perimeter includes only the solid channel or conduit boundary that is in contact with the water.
What is the hydraulic radius of a full circular pipe?
For a full circular pipe, \(R_h=D/4\). Therefore a 24 in inside-diameter pipe has a hydraulic radius of 6 in when full.
Why is a half-full circular pipe also \(R_h=D/4\)?
At half depth, both the flow area and the wetted perimeter are exactly half of the corresponding full-pipe values, so the ratio \(A/P\) remains \(D/4\).
Is hydraulic radius the same as hydraulic diameter?
No. Hydraulic diameter is four times hydraulic radius: \(D_h=4R_h=4A/P\).
Is hydraulic radius the same as hydraulic depth?
No. Hydraulic radius is \(A/P\), while hydraulic depth for an open channel is \(A/T\), where \(T\) is top width. Hydraulic depth is commonly used in Froude-number calculations.
How is hydraulic radius used in Manning’s equation?
Manning’s equation uses \(R_h^{2/3}\) together with flow area, roughness coefficient, and slope to estimate open-channel velocity or discharge under the applicable flow assumptions.
What side-slope value should I enter for a 3H:1V channel?
Enter \(z=3\). The calculator defines side slope as horizontal distance per one unit vertical, not as an angle in degrees.