Hydraulic Radius Interactive Calculator

If you don't know the hydraulic radius when sizing a storm sewer, irrigation canal, or culvert, you're basically guessing at how fast water will move and how much you can handle. That's how you end up with systems that don't work as intended. This Hydraulic Radius Calculator helps you get actual numbers for flow area, wetted perimeter, and hydraulic radius (R_h) based on your channel's shape—rectangular, trapezoidal, or circular. Getting R_h right is a core part of drainage, irrigation, or stream design because it feeds directly into Manning's equation, which tells you if your channel actually carries the required flow. Below you'll find the formulas, a worked-out canal example, basics of flow, and a practical FAQ.

What is hydraulic radius?

Hydraulic radius tells you how well a channel moves water. It's simply the cross-sectional flow area divided by the wetted perimeter—that's the part of the channel touching water. Channels with a higher hydraulic radius have less wall friction for their water volume, so the water moves easier.

Simple Explanation

Hydraulic radius boils down to this: it's the ratio of water in the channel to how much wall is in contact with that water. A wide, deep channel has far more water compared to its contact area, so water moves with less resistance. In a shallow, narrow stream, most of the water rubs along the banks and bottom, and friction slows things down. If you want water to move efficiently, you want a larger hydraulic radius.

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Cross-Section Diagram

How to Use This Calculator

1. Pick your channel shape from the dropdown (rectangular, trapezoidal, circular/partial pipe, find depth from R_h, find width from R_h, or Manning velocity).
2. Enter your known dimensions—width, depth, side slope, diameter, or hydraulic radius—depending on the mode.
3. If you're using the Manning velocity option, you'll also need channel slope and the roughness coefficient.
4. Click Calculate to get your answer.

Interactive Hydraulic Radius Calculator

Equations & Variables

Here are the base formulas for hydraulic radius for different types of channels or pipes:

Simple Example

Rectangular channel: width B = 4 m, flow depth y = 2 m.

Flow area: A = 4 × 2 = 8 m²

Wetted perimeter: P = 4 + (2 × 2) = 8 m

Hydraulic radius: R_h = 8 / 8 = 1.00 m

Theory & Practical Applications

Fundamental Concept and Physical Significance

Hydraulic radius is just flow area divided by wetted perimeter. It applies to any channel shape, not just circles. The more area you have per wetted perimeter, the less friction the flow sees from the channel boundary, which means less energy loss and higher velocity for the same slope. That's why semicircular channels have lower flow resistance than rectangles—even if both have the same area, the semicircle minimizes perimeter for that area. For open channels, hydraulic radius is the main number you'll need for most resistance and velocity calculations, including Reynolds number and both Manning and Darcy-Weisbach equations. If you have a very wide, shallow channel, the width is so large compared to the depth that hydraulic radius is basically equal to flow depth. But with narrow or deep channels, a lot more water touches the sides and bottom, so R_h is less than depth.

You'll see hydraulic radius plugged into Reynolds number (Re = VR_h/ν) and anywhere you see resistance/friction models. It's the go-to value for engineers dealing with open flow—but just keep in mind how the shape and sidewall play a big part.

Channel Geometry and Hydraulic Efficiency

Shape matters a lot: different cross sections are not equally efficient for moving water. Semicircular is best from a pure flow perspective—least wetted perimeter for a given flow area. Of course, you don't see many man-made semicircular channels because they're hard and expensive to build. Rectangular channels are easier to form with concrete but less efficient; the hydraulic radius is less than half the flow depth unless the width is almost infinite. Trapezoidal channels with side slopes you can actually build from dirt (say 1.5:1 or 2:1) provide a balance between stability and reasonable hydraulic efficiency, which is why you see them for irrigation and drainage. The "best hydraulic section" simply means the shape that gives you the biggest R_h for your constraints; for rectangles, that's width = 2 × depth, for a given trapezoid it's when half the top width matches the sloped side length, and so on. Even then, you have to consider what happens on-site: sediment build-up, how you'll maintain the channel, slope failures, and weed control can matter more in the end than maximizing R_h on paper.

Manning Equation and Flow Velocity Prediction

The Manning equation is what most engineers turn to for getting velocity from hydraulic radius. The velocity scales with R_h^2/3, so you do get more velocity as you increase R_h, but not a straight-line increase. Manning's n covers friction from the boundary surface—smooth concrete gives you low n (like 0.010), rough earth with weeds is more like 0.035. The form of Manning's equation works well enough for sizing ditches, channels, small streams, and sewers, though it assumes steady/uniform flow—meaning, depth and velocity don't change along the reach. In reality, flow often isn't uniform, so the numbers from Manning's equation only apply exactly at the section you used for your calculation. When laying out storm sewers, for instance, it's common practice to run these numbers for each pipe segment between two manholes, updating the hydraulic radius as you go based on actual predicted water depth.

Partially Full Circular Pipe Flow

Circular pipes running less than full can be counterintuitive. Maximum hydraulic radius—for maximum flow efficiency—doesn't happen when the pipe is half full but at about 81% of the diameter in depth. Why? Up to that point, the area increases faster than the perimeter; after that, the perimeter grows more quickly and starts to hurt your efficiency. That's why the maximum velocity doesn't actually happen in a full pipe, but a bit before that point. If you run a sewer pipe right at "full," you've already lost some capacity and become more prone to surcharging if you have a bit of extra flow during storms. That's why good practice is to size pipes so your peak flow hits 80–85% full, not brim-full. For depths less than half full, shortcuts won’t cut it—the full geometric calculation has to be used, or you'll introduce noticeable error.

Worked Example: Irrigation Canal Design

Let's say you need an earthen trapezoidal irrigation ditch to carry Q = 8.5 m³/s, slope S = 0.0008, max safe side slopes z = 2:1, and you plan on cutting it in earth with minimal vegetation, using Manning’s n = 0.028. The quickest way to a first pass is to use the "best hydraulic section." For a trapezoid, this happens when half the top width matches the sloped side length. Using the formulas:
Top width T = b + 2zy, sloped side = y√(1+z²), so (b + 2zy)/2 = y√(1+z²), meaning b = 2y[√(1+z²) - z]. With z = 2, b = 2y(2.236 - 2) = 0.472y.
Area: A = (b + zy)y = (0.472y + 2y)y = 2.472y².
Perimeter: P = b + 2y√(1+z²) = 0.472y + 4.472y = 4.944y.
So hydraulic radius is R_h = A/P = 0.5y. That means for the optimal trapezoid, hydraulic radius is always half the depth, whatever z is.
Plugging this into the Manning equation for Q then becomes an equation in y, and you just solve iteratively (try, check, and home in). If you find that depth gives you more Q than you need, reduce it and try again. It's rare to get a perfect match without fine-tuning y a couple of times.
Bottom line: best-section formulas save you some work, but the actual solution always comes down to plugging in values, checking Q and velocity, and adjusting depth to match your flow needs.

Critical Depth and Hydraulic Radius in Supercritical Flow

For open channels, the Froude number (Fr = V/√(gy)) tells you if flow is subcritical or supercritical, but hydraulic radius isn't the only critical dimension for these calculations—hydraulic depth D_h = A/T (T is top width) comes up too. Hydraulic radius is about resistance; hydraulic depth gets used for wave speed and critical depth calculations. At critical flow, the Froude number is 1, which is the dividing line for energy and momentum transitions. In practice, jumps (hydraulic jumps, sudden shifts from supercritical to subcritical flow) are related to these depth and area properties more than directly to R_h. For day-to-day channel work, just be clear what dimension you’re plugging in where—mixing up R_h and D_h can throw off your energy balance calculations.

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Frequently Asked Questions

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