Hydraulic Radius Calculator
Compute hydraulic radius, flow area, or wetted perimeter using the core relationship \(R = A / P\), with quick stats and detailed step-by-step calculations.
Calculation Steps
Open-channel hydraulics
Hydraulic Radius Calculator
Learn how to use the Hydraulic Radius Calculator, interpret \(R = A / P\), and connect the output to Manning’s equation, channel efficiency, and real-world design checks.
Quick Start: Using the Hydraulic Radius Calculator
The hydraulic radius \(R\) is defined as flow area divided by wetted perimeter:
\[ R = \frac{A}{P} \]
This calculator lets you solve for any one of \(R\), \(A\), or \(P\) as long as you know the other two. Follow these steps to get a reliable result and avoid the most common mistakes.
- 1 Pick what you want to solve for. In the Solve For dropdown, choose Hydraulic radius \(R\), Flow area \(A\), or Wetted perimeter \(P\). The calculator automatically hides the input you are solving for.
- 2 Enter the known values with the right units. For area, use m² or ft². For length quantities (radius and perimeter), use m, ft, or mm. The calculator converts everything internally to SI units, so mixing metric and imperial is allowed.
- 3 Use geometric formulas to get \(A\) and \(P\) if needed. For a rectangular channel, for example, \(A = b y\) and \(P = b + 2y\), where \(b\) is width and \(y\) is flow depth. For a trapezoidal section, \(P = b + 2y\sqrt{1+m^2}\) if the side slopes are \(m{:}1\) (horizontal:vertical).
- 4 Check that your numbers are physically reasonable. Flow area and wetted perimeter must be positive. Hydraulic radius should usually be less than the normal depth and certainly less than any characteristic dimension of the channel.
- 5 Interpret the result in context. A larger hydraulic radius generally means a more hydraulically efficient section. For Manning’s equation, a bigger \(R\) increases velocity and discharge for the same slope and roughness.
- 6 Review the quick stats and steps. After a valid calculation, the Quick Stats table summarizes all three variables and the hydraulic diameter \(D_h = 4R\). The Steps panel shows the algebra and substitutions used.
- 7 Save or document the result. Use the Share menu to copy a clean link, share via your device, or print a calculation sheet for your design folder.
Tip: Pair with a Manning’s equation calculator
Hydraulic radius is often just one step in your workflow. Once you have \(R\), plug it into Manning’s equation \(\bigl(V = \tfrac{1}{n}R^{2/3}S^{1/2}\bigr)\) to estimate velocity and discharge in open channels.
Warning: Don’t confuse hydraulic radius with depth
For many sections, \(R\) is smaller than the flow depth and should not be used directly as a geometric height. Using \(R\) where the depth \(y\) is required can lead to unsafe designs.
Choosing Your Method: How Are You Using Hydraulic Radius?
The Hydraulic Radius Calculator is intentionally geometry-agnostic: it works with any cross section as long as you know the flow area and wetted perimeter. In practice, engineers usually fall into one of these workflows:
1. Direct geometric calculation + hydraulic radius
You start from a known channel shape, compute \(A\) and \(P\) using geometry, and then calculate \(R = A/P\) with the calculator.
Pros
- Works for any cross section (rectangular, trapezoidal, circular, natural).
- Keeps the calculator simple and robust.
- Easy to double-check with hand calculations.
Cons
- Requires you to know or compute \(A\) and \(P\) accurately.
- For irregular natural channels, estimating \(A\) and \(P\) can be tedious.
2. Back-calculating area or perimeter from target \(R\)
Sometimes you know the hydraulic radius you want (from a standard section or a target efficiency) and need to solve for flow area or wetted perimeter consistent with that value.
Pros
- Useful when matching a published channel template or code requirement.
- Helps you understand how much perimeter you can “afford” for a given area.
Cons
- Still requires a geometric model to translate \(A\) and \(P\) into actual dimensions.
- Not a substitute for full hydraulic checks (freeboard, velocities, shear stress).
3. Field measurements & as-built checks
In existing channels, you may measure water depth and cross-section shape in the field, estimate \(A\) and \(P\), and use the calculator to back out the actual operating hydraulic radius.
Pros
- Grounds your model in real-world measurements.
- Helps calibrate roughness coefficients and verify capacity.
Cons
- Measurements can be noisy, especially in natural streams.
- Requires careful documentation of stationing and cross-section geometry.
Tip: Keep geometry and hydraulics separated
Use this Hydraulic Radius Calculator purely for the \(R = A/P\) relationship, and let other tools handle geometric optimization or energy-grade-line analysis. This separation makes debugging much easier.
What Moves the Number: Levers Behind Hydraulic Radius
For a fixed discharge and slope, hydraulic radius is one of the main levers controlling velocity and depth (via Manning’s equation). Even though the calculator focuses on \(R\), understanding what changes it helps you design better channels.
Increase flow area \(A\)
For the same wetted perimeter \(P\), a larger cross-sectional area raises \(R\). Widening a channel or increasing flow depth can both increase \(A\), improving hydraulic efficiency.
Reduce wetted perimeter \(P\)
For the same area, reducing the boundary in contact with water (e.g., smoother lining, fewer ledges, rounded corners) increases \(R\) and typically reduces head losses.
Channel shape optimization
Among sections with the same area, a semi-circular shape maximizes hydraulic radius, while very wide, shallow rectangles tend to have lower \(R\). Trapezoidal sections offer a practical balance in earth channels.
Partial filling in closed conduits
For circular pipes, hydraulic radius changes significantly as the pipe goes from low flow to full flow. The same physical conduit can operate with very different \(R\) depending on depth.
Roughness still matters
A higher hydraulic radius does not guarantee lower energy losses if the roughness coefficient \(n\) is large. A rough rock-lined channel with large \(R\) may still have lower velocity than a smooth concrete channel with slightly smaller \(R\).
Scale and units
Because \(R\) is in units of length (m or ft), scaling the entire cross section by a constant factor scales \(R\) by the same factor. Make sure the calculator inputs and your hand checks use consistent units.
Worked Examples: Hydraulic Radius in Practice
These examples use realistic dimensions that you can replicate in the Hydraulic Radius Calculator. Adjust the values to match your own project, then compare the outputs.
Example 1 — Rectangular concrete channel, solve for hydraulic radius
Given
- Channel width \(b = 3.0\ \text{m}\)
- Flow depth \(y = 1.2\ \text{m}\)
- Rectangular section, smooth concrete
Compute flow area \(A\)
For a rectangular section:
\[ A = b y = 3.0 \times 1.2 = 3.6\ \text{m}^2 \]
Compute wetted perimeter \(P\)
Wetted perimeter includes the bottom and two side walls:
\[ P = b + 2y = 3.0 + 2(1.2) = 5.4\ \text{m} \]
Compute hydraulic radius \(R\)
Now compute:
\[ R = \frac{A}{P} = \frac{3.6}{5.4} = 0.667\ \text{m} \]
In the calculator, choose “Solve for hydraulic radius \(R\)”, enter \(A = 3.6\ \text{m}^2\), \(P = 5.4\ \text{m}\), and confirm that the output matches \(R \approx 0.667\ \text{m}\).
Example 2 — Trapezoidal earth channel, solve for flow area
Given
- Bottom width \(b = 4.0\ \text{m}\)
- Side slopes \(m = 2\) (horizontal:vertical)
- Flow depth \(y = 1.0\ \text{m}\)
- Target hydraulic radius \(R = 0.9\ \text{m}\)
Compute trapezoidal area \(A\)
The top width is:
\[ T = b + 2my = 4.0 + 2(2)(1.0) = 8.0\ \text{m} \]
So the area is:
\[ A = \frac{(b + T)}{2} y = \frac{(4.0 + 8.0)}{2} (1.0) = 6.0\ \text{m}^2 \]
Compute wetted perimeter \(P\)
Each side length is:
\[ L_s = \sqrt{(my)^2 + y^2} = \sqrt{(2 \cdot 1.0)^2 + 1.0^2} = \sqrt{5} \approx 2.236\ \text{m} \]
So:
\[ P = b + 2L_s = 4.0 + 2(2.236) \approx 8.472\ \text{m} \]
Check the resulting hydraulic radius
Using the calculator with \(A = 6.0\ \text{m}^2\) and \(P = 8.472\ \text{m}\):
\[ R = \frac{6.0}{8.472} \approx 0.708\ \text{m} \]
Since this is below the target \(R = 0.9\ \text{m}\), you would need to adjust geometry (e.g., increase depth or bottom width) and recompute until the calculated \(R\) matches your design goal.
Example 3 — Circular storm sewer, wetted perimeter at half full
Given
- Pipe diameter \(D = 1.2\ \text{m}\)
- Flow depth \(y = 0.6\ \text{m}\) (half full)
- Known hydraulic radius at half full \(R \approx 0.30\ \text{m}\)
Estimate flow area \(A\)
At exactly half full, the flow area is half the full cross-sectional area:
\[ A = \tfrac{1}{2}\left(\frac{\pi D^2}{4}\right) = \tfrac{1}{2}\left(\frac{\pi (1.2)^2}{4}\right) \approx 0.565\ \text{m}^2 \]
Solve for wetted perimeter \(P\)
Using the definition \(P = A / R\):
\[ P = \frac{A}{R} = \frac{0.565}{0.30} \approx 1.88\ \text{m} \]
In the calculator, choose “Solve for wetted perimeter \(P\)”, enter \(A = 0.565\ \text{m}^2\), \(R = 0.30\ \text{m}\), and confirm that the computed \(P\) is about \(1.88\ \text{m}\).
Common Layouts & Variations
Different channel shapes and materials lead to very different hydraulic radii, even for the same discharge. Use the table below as a qualitative guide when interpreting calculator results.
| Layout | Typical application | Hydraulic radius behavior | Pros | Cons |
|---|---|---|---|---|
| Rectangular concrete channel | Urban drainage, treatment plant channels | Moderate \(R\); increases with depth until nearly full. Corners reduce efficiency compared to curved sections. | Straightforward geometry, easy to form and inspect. | Higher wetted perimeter for the same area than curved sections. |
| Trapezoidal earth channel | Irrigation canals, roadside ditches | \(R\) depends on side slopes and depth. Flatter side slopes increase area faster than perimeter, improving \(R\), but require more right-of-way. | Constructible with earthmoving, stable side slopes. | Roughness can be high; sediment and vegetation reduce effective area. |
| Circular pipe (part full) | Storm sewers, sanitary sewers | \(R\) varies with depth; reaches a maximum at a specific filling ratio. At very shallow depths, \(R\) is small and capacity is limited. | Very efficient when sized correctly; strong structural shape. | Capacity is sensitive to depth; surcharge behavior must be checked. |
| Natural stream, irregular section | Rivers, creeks, restored channels | Effective \(R\) can be low due to rough banks and bed forms. Cross sections often require surveying or 2D modelling. | Can provide ecological benefits and bank stability. | Highly variable geometry; difficult to characterize with a single \(R\). |
| Engineered “best hydraulic” section | Special channels near capacity limits | Shapes are tuned to maximize \(R\) for a given area, often approaching circular or parabolic forms. | Excellent hydraulic efficiency and compact footprint. | More complex forms and construction details; higher initial cost. |
Tip: Compare \(R\) between alternatives
When choosing between two layouts, compute \(R\) for each at the design depth. A seemingly minor geometry tweak can produce a noticeable change in hydraulic radius and therefore required slope.
Specs, Logistics & Sanity Checks
Hydraulic radius by itself does not guarantee a safe or code-compliant design. Use the calculator as part of a broader workflow that includes roughness selection, freeboard, and structural checks.
Before you finalize a design
- Verify units: keep a consistent set (m–m²–m³/s or ft–ft²–cfs) throughout your calculations.
- Confirm geometry: ensure the \(A\) and \(P\) values input to the calculator match your latest cross-section sketch or CAD model.
- Check freeboard: hydraulic radius is not the same as water depth. Always check that the water surface is safely below the top of the section.
- Run a capacity check: combine \(R\) with Manning’s equation or a more advanced model to verify discharge at design storm conditions.
- Consider sediment and debris: channels can lose effective area over time due to deposition and vegetation.
- Document assumptions: note which roughness \(n\), slope, and geometry you assumed when you used the calculator.
Field and construction notes
- Survey as-built sections for critical channels and recalculate \(R\) to compare against design values.
- For lined channels, confirm the lining thickness doesn’t materially change area and perimeter compared to the design section.
- In natural streams, consider using several cross sections and average \(R\) where appropriate.
- Maintain channels: vegetation management, sediment removal, and bank protection can preserve both \(A\) and \(R\) over time.
Warning: Don’t design by hydraulic radius alone
A channel with a high hydraulic radius but excessive velocity may cause erosion or scour. Always combine hydraulic radius calculations with shear stress checks, velocity limits, and local design standards.
Frequently Asked Questions
What is hydraulic radius and why is it important?
Hydraulic radius \(R\) is defined as the ratio of flow area to wetted perimeter:
\[ R = \frac{A}{P} \]
It is a geometric indicator of how efficiently a channel conveys flow. For a given slope and roughness, channels with larger \(R\) generally have higher velocities and greater capacity. Hydraulic radius is a key input to empirical formulas such as Manning’s equation.
How is hydraulic radius used in Manning’s equation?
In Manning’s equation for open-channel flow,
\[ V = \frac{1}{n} R^{2/3} S^{1/2} \]
the hydraulic radius \(R\) is raised to the power of \(2/3\). A larger \(R\) increases the calculated velocity \(V\) for a given slope \(S\) and roughness coefficient \(n\). Once velocity is known, discharge follows as \(Q = A V\).
Is hydraulic radius the same as flow depth?
No. Hydraulic radius is not the same as flow depth, even though both have units of length. For many sections (such as rectangles and trapezoids), \(R\) is smaller than the depth. Using hydraulic radius where depth is required can lead to incorrect freeboard, velocity, and stability checks.
Can I mix metric and imperial units in the calculator?
Yes. The calculator converts all inputs to SI units internally, so you can enter lengths in feet and areas in m² if needed. However, it is best practice to keep a consistent unit system within your broader design calculations to avoid transcription errors.
How accurate is hydraulic radius for natural streams?
In natural streams with irregular banks, vegetation, and bed forms, a single hydraulic radius is an approximation. It is still useful for capacity and flood studies, but it should be based on good survey data and supplemented with engineering judgment, multiple cross sections, and possibly 2D modelling for complex reaches.
What is hydraulic diameter and how does it relate to hydraulic radius?
Hydraulic diameter \(D_h\) is defined as four times the hydraulic radius:
\[ D_h = 4R \]
It is commonly used in pipe-flow correlations and dimensionless numbers such as Reynolds number. The calculator’s quick stats section automatically reports \(D_h\) once \(R\) is known.
What are common mistakes when using a hydraulic radius calculator?
Common mistakes include mixing units without realizing it, using the water surface width as wetted perimeter, entering geometric dimensions instead of the derived \(A\) and \(P\), and assuming \(R\) is equal to depth. Always sketch the cross section, compute \(A\) and \(P\) carefully, and then use the calculator to perform the final ratio.