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.
On this page
Calculator Guide
How to Use the Hydraulic Radius Calculator
The Hydraulic Radius Calculator above calculates hydraulic radius from either known flow area and wetted perimeter or from common channel and pipe geometry. Hydraulic radius is the cross-sectional flow area divided by the wetted perimeter, and it is commonly used in open-channel flow, Manning’s equation, drainage ditches, canals, culverts, and partially full pipe calculations.
Use the calculator when you need a fast geometry check for a rectangular channel, trapezoidal channel, triangular channel, full circular pipe, partially full circular pipe, or a custom section where area and wetted perimeter are already known. The key is entering the wetted perimeter correctly: in open channels, the free water surface is not part of the wetted perimeter.
Quick Answer
Hydraulic radius is calculated with \(R_h=A/P\), where \(A\) is the flow area and \(P\) is the wetted perimeter. For a rectangular channel, \(R_h=by/(b+2y)\). For a full circular pipe, \(R_h=D/4\). For a partially full pipe, the area and wetted arc depend on water depth and central angle.
Do not rely on this simplified result when…
Do not use hydraulic radius alone as a final drainage design. Hydraulic radius describes geometry, but final open-channel or culvert design also depends on slope, roughness, flow rate, inlet/outlet conditions, sediment, freeboard, tailwater, local criteria, and professional engineering judgment.
Inputs and Outputs Used by the Calculator
The calculator changes its required inputs based on the geometry mode. If you already know flow area and wetted perimeter, use the custom mode. If you know physical dimensions, choose the matching channel or pipe shape.
| Type | Value | What It Means | Common Unit |
|---|---|---|---|
| Input | Flow Area, \(A\) | Cross-sectional area occupied by flowing water. | ft², m², in² |
| Input | Wetted Perimeter, \(P\) | Length of solid boundary in contact with water. | ft, m, in |
| Input | Bottom Width, \(b\) | Flat bottom width for rectangular or trapezoidal channels. | ft, m, in |
| Input | Flow Depth, \(y\) | Vertical depth of water in the channel or pipe. | ft, m, in |
| Input | Side Slope, \(z:1\) | Horizontal-to-vertical side slope for trapezoidal or triangular channels. | dimensionless |
| Input | Pipe Diameter, \(D\) | Inside diameter of a full or partially full circular pipe. | in, ft, m |
| Output | Hydraulic Radius, \(R_h\) | Flow area divided by wetted perimeter. | ft, m, in |
| Output | Hydraulic Diameter, \(D_h\) | Four times the hydraulic radius. | ft, m, in |
| Output | Central Angle, \(\theta\) | Angle used to calculate a partially full circular pipe segment. | radians |
| Output | Top Width and Hydraulic Depth | Open-channel checks used when a free surface is present. | ft, m, in |
Hydraulic Radius Formula
Hydraulic radius is the ratio of flow area to wetted perimeter. Because area divided by length produces length, hydraulic radius is reported in feet, meters, inches, centimeters, or another length unit.
Main Formula
\(R_h\) is hydraulic radius, \(A\) is flow area, and \(P\) is wetted perimeter.
Hydraulic Diameter
Hydraulic diameter is four times hydraulic radius. It is not two times hydraulic radius.
What Is Wetted Perimeter?
Wetted perimeter is the length of channel or pipe boundary in direct contact with water. In an open channel, the free water surface is not included because it is not touching the channel bed or walls. In a partially full pipe, the wetted perimeter is the wetted arc length, not the full pipe circumference unless the pipe is full.
Important wetted perimeter rule
If a surface is water-to-air, it is not part of the wetted perimeter. If a surface is water-to-solid boundary, it is part of the wetted perimeter.
| Shape | Flow Area | Wetted Perimeter | Hydraulic Radius |
|---|---|---|---|
| Rectangular channel | \(A=by\) | \(P=b+2y\) | \(R_h=\frac{by}{b+2y}\) |
| Trapezoidal channel | \(A=y(b+zy)\) | \(P=b+2y\sqrt{1+z^2}\) | \(R_h=\frac{A}{P}\) |
| Triangular channel | \(A=zy^2\) | \(P=2y\sqrt{1+z^2}\) | \(R_h=\frac{A}{P}\) |
| Full circular pipe | \(A=\frac{\pi D^2}{4}\) | \(P=\pi D\) | \(R_h=\frac{D}{4}\) |
| Partially full circular pipe | \(A=\frac{r^2}{2}(\theta-\sin\theta)\) | \(P=r\theta\) | \(R_h=\frac{A}{P}\) |
Rectangular Channel
Trapezoidal Channel
Triangular Channel
A trapezoidal channel with \(b=0\) behaves like a triangular V-channel.
Full Circular Pipe
Partially Full Circular Pipe
The angle \(\theta\) must be in radians. When \(y=D\), \(\theta=2\pi\) and the formula becomes the full-pipe case. When \(y=D/2\), \(\theta=\pi\), and the hydraulic radius is also \(D/4\).
What the Variables Mean
The variables depend on the selected shape. The most important distinction is between total perimeter and wetted perimeter. Only the boundary touching water counts as wetted perimeter.
| Symbol | Meaning | How to Enter It |
|---|---|---|
| \(R_h\) | Hydraulic radius. | The calculator returns this as the primary result. |
| \(A\) | Cross-sectional flow area. | Enter the area occupied by water, not necessarily the full channel or pipe area. |
| \(P\) | Wetted perimeter. | Enter only the solid boundary length in contact with water. |
| \(b\) | Bottom width of a rectangular or trapezoidal channel. | Use the flat bottom width, not the top water surface width. |
| \(y\) | Flow depth. | Use vertical water depth from the channel bottom or pipe invert. |
| \(z\) | Side slope ratio. | Enter horizontal-to-vertical slope. A 3:1 slope is entered as \(z=3\). |
| \(D\) | Pipe diameter. | Use inside pipe diameter for circular pipe calculations. |
| \(r\) | Pipe radius. | \(r=D/2\). The calculator derives this from diameter. |
| \(\theta\) | Central angle for a partially full pipe. | Calculated internally in radians from pipe diameter and flow depth. |
How to Use the Calculator
Start by choosing the geometry that matches your section. Then enter the required dimensions, select units, and check the area, wetted perimeter, and hydraulic radius result.
Choose the geometry type
Use custom area and wetted perimeter, rectangular channel, trapezoidal channel, triangular channel, full circular pipe, or partially full circular pipe.
Enter the known values
Enter dimensions such as bottom width, flow depth, side slope, pipe diameter, or known area and wetted perimeter.
Check the wetted perimeter
For open channels, do not include the water surface. For partially full pipes, use only the wetted arc, not the full circumference.
Review the supporting outputs
Use area, wetted perimeter, hydraulic diameter, top width, hydraulic depth, percent full, and central angle to confirm the result makes sense.
| If You Know… | Use This Mode | Main Caution |
|---|---|---|
| Flow area and wetted perimeter | Custom Area & Wetted Perimeter | Make sure \(P\) excludes the free surface. |
| Bottom width and flow depth | Rectangular Channel | Use vertical depth, not sloped side length. |
| Bottom width, depth, and side slope | Trapezoidal Channel | Enter side slope as \(z:1\), not degrees. |
| V-channel depth and side slope | Triangular Channel | Use symmetric side slopes unless your section requires custom geometry. |
| Pipe diameter and pipe is full | Full Circular Pipe | Use inside diameter. |
| Pipe diameter and water depth | Partially Full Circular Pipe | Depth must be greater than 0 and no greater than \(D\). |
How to Interpret Hydraulic Radius Results
Hydraulic radius represents how much flow area is available per unit of wetted boundary. A larger value usually indicates a more hydraulically efficient section, but it does not by itself determine flow rate.
| Result Pattern | What It May Mean | What to Check Next |
|---|---|---|
| Very small \(R_h\) | Small flow area relative to boundary contact. Shallow flow often produces small values. | Check depth, wetted perimeter, and units. |
| \(R_h\) approaches flow depth in a wide channel | This can be reasonable for a very wide rectangular channel because the side walls contribute little to total wetted perimeter. | Check whether the channel width is much larger than the flow depth. |
| Full circular pipe gives \(R_h=D/4\) | This is the expected relationship for a full pipe. | Confirm inside diameter and output units. |
| Half-full circular pipe gives \(R_h=D/4\) | Both area and wetted perimeter are half of the full-pipe values, so their ratio remains the same. | Use this as a useful benchmark for partial-pipe calculations. |
| \(R_h\) larger than expected | Wetted perimeter may be too small or area may have been entered in the wrong units. | Recalculate \(A\) and \(P\) from geometry. |
| Zero, negative, or impossible result | Input area, depth, diameter, or wetted perimeter is invalid. | All physical dimensions must be positive, except trapezoid bottom width may be zero. |
What to do with the result
Use hydraulic radius as a geometry input for open-channel flow checks such as Manning’s equation. To estimate velocity or discharge, you still need channel slope, roughness coefficient, and flow area.
What changes the result most?
Flow depth usually has a major effect because it changes both area and wetted perimeter. In channels, increasing depth often increases \(A\) faster than \(P\), which can increase \(R_h\). In partially full pipes, the relationship is nonlinear because both water area and wetted arc depend on the circular segment angle.
Quick sanity check
For a full circular pipe, the result should equal \(D/4\). For a rectangular channel, \(R_h\) should be less than the flow depth \(y\), because \(R_h=by/(b+2y)\). If the calculated hydraulic radius is larger than the flow depth for a rectangular channel, check the wetted perimeter and units.
Input Quality Checklist
Most wrong hydraulic radius results come from incorrect wetted perimeter, inconsistent units, or using the wrong shape mode. Check these items before trusting the output.
Wetted perimeter
Include only solid surfaces touching water. Do not include the open water surface in an open channel.
Inside pipe diameter
For circular pipes, use the inside diameter, not the outside diameter or nominal size unless they match your design assumption.
Side slope format
Enter \(z\) as horizontal-to-vertical. A 3H:1V side slope is entered as 3, not as an angle in degrees.
Depth limit
For a partially full pipe, water depth must satisfy \(0<y\le D\). Depth greater than diameter is not physically valid.
Unit consistency
Do not mix inches, feet, meters, and square units without conversion. Area units and length units convert differently.
Shape assumption
If the channel has benches, compound sections, irregular sides, or rough natural banks, a simple shape mode may be too idealized.
Step-by-Step Worked Examples
The examples below show how hydraulic radius is calculated for the most common user cases: a rectangular channel, a trapezoidal drainage channel, and a circular pipe.
Calculate Flow Area
Calculate Wetted Perimeter
Calculate Hydraulic Radius
Final Answer
Hydraulic radius: \(R_h=1.2\,ft\). This is reasonable because the section has a moderate flow area relative to the bottom and two wetted side walls.
Calculate Flow Area
Calculate Wetted Perimeter
Calculate Hydraulic Radius
Final Answer
Hydraulic radius: \(R_h\approx1.20\,ft\). This is a useful benchmark for checking side-slope inputs in trapezoidal ditch or canal calculations.
Use the Full-Pipe Relationship
Verify from Area and Perimeter
Final Answer
Hydraulic radius: \(R_h=6\,in\). This confirms the full-pipe rule \(R_h=D/4\).
Half-full pipe insight
A half-full circular pipe also gives \(R_h=D/4\). Both area and wetted perimeter are half of the full-pipe values, so the ratio \(A/P\) remains the same. This is a useful check for partially full pipe calculations.
Engineering Diagram
The most important visual concept is the difference between wetted perimeter and free water surface. The wetted perimeter follows the solid channel or pipe boundary touching water; it does not follow the open water surface.
Reference Values and Reasonableness Checks
Hydraulic radius does not have one universal “good” value because it depends on channel size and shape. Instead, compare the result to the section dimensions and known shape relationships.
| Shape or Case | Useful Check | Why It Helps |
|---|---|---|
| Full circular pipe | \(R_h=D/4\) | Simple exact relationship for a full pipe. |
| Half-full circular pipe | \(R_h=D/4\) | A half-full circular pipe has the same hydraulic radius as a full pipe, even though the area and perimeter are half as large. |
| Wide rectangular channel | \(R_h\) approaches \(y\) | When width is much larger than depth, side walls matter less. |
| Very shallow flow | \(R_h\) is usually small | Shallow flow has relatively high boundary contact compared with area. |
| Trapezoidal channel with \(b=0\) | Behaves like triangular channel | Zero bottom width converts the trapezoid into a V-shaped section. |
Design Ranges and Practical Engineering Checks
Hydraulic radius is a geometry value, not a complete design result. A mathematically correct hydraulic radius still needs to be checked against flow capacity, slope, roughness, freeboard, erosion, sediment transport, and local design requirements.
Geometry Efficiency
Larger \(R_h\) usually means less wetted boundary per unit of area, which can improve hydraulic efficiency.
Manning’s Equation
Hydraulic radius is used with Manning’s \(n\), slope, and area to estimate open-channel velocity and discharge.
Field Conditions
Vegetation, sediment, debris, irregular banks, and roughness can dominate flow performance even if geometry looks efficient.
Manning Velocity Form
This SI form estimates average velocity \(V\) from roughness \(n\), hydraulic radius \(R_h\), and slope \(S\).
Manning Discharge Form
In U.S. customary units, the common Manning form uses \(Q=\frac{1.49}{n}AR_h^{2/3}S^{1/2}\).
When the result is not enough
For final drainage, culvert, stormwater, or channel design, hydraulic radius should be used as one input in a larger analysis. It does not by itself confirm capacity, stability, freeboard, backwater effects, or code compliance.
Unit Conversion Notes
Hydraulic radius has units of length. Area inputs must be converted as square units, while perimeter and dimension inputs must be converted as length units.
| Quantity | Common Units | Conversion Reminder |
|---|---|---|
| Length | ft, in, m, cm, mm | \(1\,ft=0.3048\,m\), \(1\,in=0.0254\,m\) |
| Area | ft², in², m², cm², mm² | \(1\,ft^2=0.09290304\,m^2\), \(1\,in^2=0.00064516\,m^2\) |
| Side slope | \(z:1\) | A 3H:1V slope is entered as \(z=3\), not as 3 degrees. |
| Partial pipe angle | radians | Circular segment formulas use radians internally. |
Common unit trap
Do not convert area with the same factor as length. For example, \(1\,ft=0.3048\,m\), but \(1\,ft^2=0.09290304\,m^2\). Area conversion factors are squared length conversion factors.
Hydraulic Radius vs. Related Hydraulic Terms
Hydraulic radius is often confused with hydraulic diameter, hydraulic depth, and pipe radius. These terms are related, but they are not interchangeable.
| Term | Formula | Main Use |
|---|---|---|
| Hydraulic Radius | \(R_h=A/P\) | Open-channel and conduit geometry efficiency. |
| Hydraulic Diameter | \(D_h=4R_h=4A/P\) | Equivalent diameter for non-circular flow sections. |
| Hydraulic Depth | \(D_{hyd}=A/T\) | Open-channel free-surface flow calculations such as Froude number. |
| Pipe Radius | \(r=D/2\) | Geometric radius of a circular pipe. |
| Manning Flow Calculation | \(V=\frac{1}{n}R_h^{2/3}S^{1/2}\) | Velocity estimate from roughness, slope, and hydraulic radius. |
Key difference
Hydraulic diameter is four times hydraulic radius. Pipe radius is a geometric dimension. For a full circular pipe, \(R_h=D/4=r/2\), so hydraulic radius is not the same as pipe radius.
Common Mistakes That Cause Wrong Hydraulic Radius Results
Most errors come from using the wrong perimeter, the wrong shape assumption, or inconsistent units.
Common Mistakes
- Including the open water surface in wetted perimeter.
- Using total pipe circumference for a partially full pipe.
- Confusing hydraulic radius with pipe radius.
- Entering side slope as degrees instead of horizontal-to-vertical ratio.
- Mixing inches and feet without converting area and length separately.
- Using outside pipe diameter instead of inside pipe diameter.
Better Practice
- Trace only the boundary that water actually touches.
- Use the wetted arc for partially full pipe flow.
- Remember that full-pipe \(R_h=D/4\).
- Enter a 3:1 side slope as \(z=3\).
- Convert all dimensions to a consistent unit system before hand calculations.
- Use inside diameter for pipe flow geometry.
Troubleshooting Unexpected Results
If the hydraulic radius result looks wrong, check the geometry inputs before assuming the formula is wrong. The formula is simple; the perimeter definition is usually the problem.
| Problem | Likely Cause | Fix |
|---|---|---|
| Result is much too small | Wetted perimeter is too large, or water surface was included. | Recalculate \(P\) using only solid wetted boundary. |
| Result is much too large | Wetted perimeter is too small, or area was entered in the wrong units. | Check area units and verify boundary length. |
| Full pipe does not equal \(D/4\) | Diameter or output unit is wrong. | Use inside diameter and check unit selectors. |
| Partial pipe input fails | Flow depth exceeds pipe diameter. | Use \(0<y\le D\). |
| Trapezoid result looks odd | Side slope was entered as an angle or inverted ratio. | Enter horizontal-to-vertical slope. For 3H:1V, enter 3. |
| Math is valid but design seems unrealistic | Simple shape does not represent the real channel. | Use surveyed geometry or a more detailed hydraulic model for irregular sections. |
Common edge cases
Very shallow partial-pipe flow, nearly full pipe flow, zero-bottom-width trapezoids, and natural channels with irregular banks can all produce valid calculations that still need engineering interpretation.
Assumptions, Sources, and Limitations
This calculator is intended for geometry-based hydraulic radius calculations. It does not estimate flow rate, velocity, backwater, energy loss, channel stability, erosion, sediment movement, or code compliance by itself.
Geometry Assumption
The selected shape is assumed to represent the actual flow section. Irregular channels may require surveyed cross-section analysis.
Open-Channel Assumption
For open channels, the water surface is not counted as wetted perimeter.
Pipe Assumption
Circular pipe calculations assume an inside circular section and geometric water depth measured from the pipe invert.
Final Design Note
For final stormwater, culvert, drainage, or open-channel design, verify slope, roughness, flow rate, freeboard, tailwater, erosion, and applicable local criteria.
Calculation basis
The calculation basis is the standard hydraulic radius relationship \(R_h=A/P\), where \(A\) is cross-sectional flow area and \(P\) is wetted perimeter. The Federal Highway Administration’s Hydraulic Design Series No. 4 discusses hydraulic radius in the context of open-channel flow resistance relationships such as Manning’s equation and Darcy-Weisbach. Use project-specific standards and professional judgment for final design.
Glossary of Terms
These terms appear often in hydraulic radius, open-channel flow, pipe flow, and Manning equation calculations.
Hydraulic Radius
Flow area divided by wetted perimeter. It has units of length.
Wetted Perimeter
The length of solid boundary that is in contact with flowing water.
Flow Area
The cross-sectional area occupied by water in the channel or pipe.
Hydraulic Diameter
Four times hydraulic radius, commonly written as \(D_h=4R_h\).
Hydraulic Depth
Flow area divided by top width, usually used in open-channel free-surface calculations.
Side Slope
The horizontal-to-vertical slope ratio of a channel side, such as 3H:1V.
Central Angle
The angle used to define the wetted circular segment in a partially full pipe.
Manning’s Equation
An empirical open-channel flow equation that uses hydraulic radius, slope, and roughness.
Frequently Asked Questions
What does the Hydraulic Radius Calculator calculate?
The calculator finds hydraulic radius from flow area and wetted perimeter or from common channel and pipe shapes. It can also show supporting values such as flow area, wetted perimeter, hydraulic diameter, top width, hydraulic depth, percent full, and central angle when applicable.
What is the hydraulic radius formula?
The hydraulic radius formula is \(R_h=A/P\), where \(A\) is cross-sectional flow area and \(P\) is wetted perimeter.
Does wetted perimeter include the water surface?
No. In open-channel flow, the free water surface is not part of the wetted perimeter. Only the solid boundary touching water is included.
What is the hydraulic radius of a full circular pipe?
For a full circular pipe, \(R_h=D/4\). A 24-inch full pipe therefore has a hydraulic radius of 6 inches.
Is hydraulic radius the same as pipe radius?
No. Pipe radius is \(r=D/2\). For a full circular pipe, hydraulic radius is \(R_h=D/4=r/2\).
How is hydraulic radius used in Manning’s equation?
Manning’s equation uses hydraulic radius as the geometry term for estimating open-channel velocity or discharge. To calculate flow, you also need slope, roughness coefficient, and flow area.