The hydraulic radius calculator above turns a cross-section into the two length scales that fluid mechanics actually uses: the hydraulic radius R = A/P, which appears in Manning's equation and every open-channel formula, and the hydraulic diameter Dh = 4R, which is what you put into a Reynolds number or a friction-factor chart when the duct is not round.
Arb Digital publishes free physics calculators that each own one quantity, and four of them share the word "hydraulic" while meaning four unrelated things. This page owns A divided by P, a purely geometric property of a cross-section. The hydraulic jump calculator owns an event in a channel, the supercritical-to-subcritical transition with its sequent depth and energy loss. The hydraulic conductivity calculator owns K, how readily water moves through soil or rock. The hydraulic gradient calculator owns i, the dimensionless slope that drives groundwater flow. And the live hydraulic cylinder force calculator shares only the word: it is fluid power, turning bore and pressure into actuator force.
What This Hydraulic Radius Calculator Does
It handles six cross-sections: trapezoidal, rectangular and triangular open channels, a circular pipe both part full and running full, and a rectangular closed duct. For each it returns the flow area, the wetted perimeter, the hydraulic radius, the hydraulic diameter and R raised to the power two thirds, which is the exact factor Manning's equation needs.
The part-full circular case is where most of the difficulty lives, because the area and perimeter both involve the subtended angle rather than simple straight edges, and neither varies linearly with depth. The tool solves that geometry properly rather than approximating it.
The closed-duct option exists because the same ratio does a completely different job there. In a non-circular duct running full, the hydraulic diameter is what makes an ordinary pipe-flow correlation usable: you substitute 4R for the diameter and the Reynolds number and friction factor behave approximately as they would in a round pipe. The open channel flow calculator uses a hydraulic radius internally to solve the Manning discharge problem for a channel; this page is the geometry on its own, including the closed-duct case that open-channel work never touches.
How to Use It
- Pick the cross-section. The fields change to match, so only the dimensions that matter for that shape are shown.
- Enter the flow depth, not the channel depth. Hydraulic radius depends on how deep the water actually is, not on how deep the channel was dug.
- For a part-full pipe, keep the depth below the diameter. Above it the pipe is surcharged and behaves as a closed conduit, which is the "running full" option instead.
- Read the hydraulic diameter if you are heading for a Reynolds number. It is four times the hydraulic radius, and using R where Dh is meant is wrong by exactly that factor of four.
- Take R to the power two thirds straight into Manning's equation. That is the form the equation uses, and having it computed saves an error-prone step.
The Formula: How Hydraulic Radius Is Calculated
The definition is simply
R = A ÷ P
where A is the cross-sectional area of the flow and P is the wetted perimeter, meaning the length of the boundary actually in contact with the fluid. In an open channel the free surface is not part of the wetted perimeter, because there is no wall there to exert shear.
For a rectangular channel of bed width b at depth y, A = by and P = b + 2y. For a trapezoid with side slope z horizontal to one vertical, A = (b + zy)y and P = b + 2y√(1 + z²). For a triangular channel, A = zy² and P = 2y√(1 + z²). For a circular pipe of diameter D flowing at depth y, with θ = 2 arccos(1 − 2y/D), the area is D²(θ − sin θ)/8 and the wetted perimeter is Dθ/2. For a rectangular closed duct of width w and height h, all four walls are wetted, so P = 2(w + h) and the hydraulic diameter reduces to 2wh/(w + h).
The Bureau of Reclamation's Water Measurement Manual is the standard open reference for open-channel geometry and measurement, and MIT OpenCourseWare's Transport Processes in the Environment, from Civil and Environmental Engineering, covers the shear-stress reasoning behind why area over wetted perimeter is the length scale that matters.
Work the default by hand. A trapezoidal channel with a 3 m bed, 1.2 m of water and 2:1 side slopes has A = (3 + 2 × 1.2) × 1.2 = 5.4 × 1.2 = 6.48 m². The wetted perimeter is 3 + 2 × 1.2 × √5 = 3 + 2.4 × 2.23607 = 8.3666 m. So R = 6.48 ÷ 8.3666 = 0.7745 m, the hydraulic diameter is 3.0980 m, and R2/3 = 0.8434.
Why It Is Called A Radius When It Is Not One
The name is a historical accident and it misleads people constantly. Hydraulic radius is not half of anything. For a circular pipe running full, A = πD²/4 and P = πD, so R = D/4 — a quarter of the diameter, which is half the geometric radius.
That factor of two is precisely why the hydraulic diameter is defined as 4R rather than 2R. The definition is chosen so that for a round pipe the hydraulic diameter equals the actual diameter, which lets every existing pipe-flow correlation carry over unchanged to other shapes. Define it as 2R and every friction chart in the world would need rewriting.
The practical consequence is a very common error: substituting R into a Reynolds number where Dh belongs. That understates the Reynolds number by a factor of four, which can put a genuinely turbulent flow into the laminar band on a Moody chart and give a friction factor that is badly wrong. The Reynolds number calculator and the pipe flow calculator both expect the hydraulic diameter, not the radius.
The Part-Full Pipe Is Deeply Counter-Intuitive
A pipe running exactly half full has the same hydraulic radius as one running completely full. Both come to D/4. Run the half-full preset and then the full option to see it: area and wetted perimeter both halve, and the ratio is unchanged.
Stranger still, the hydraulic radius does not peak when the pipe is full. It rises with depth up to about 81 per cent full, where it reaches roughly 0.30D, and then falls back to 0.25D as the last of the crown is submerged. The reason is that in the top of a pipe the wetted perimeter grows quickly while the added area is a thin sliver, so the ratio deteriorates.
Because velocity in Manning's equation goes as R2/3, the mean velocity peaks at that same 81 per cent depth rather than when full. And because discharge is velocity times area, discharge peaks somewhere else again, at about 94 per cent full — a part-full sewer can carry slightly more than the same pipe running full. This is why sewer design curves are drawn against relative depth rather than assuming full-bore behaviour throughout, and it is a genuine result rather than a quirk of the approximation.
Where The Ratio Comes From, And Where It Breaks
Hydraulic radius is not an arbitrary convenience. Consider a length of channel in steady uniform flow: the weight component driving the water downhill is proportional to the area, and the boundary shear resisting it acts over the wetted perimeter. Setting the two in balance gives an average boundary shear stress of ρgRS, where S is the slope. The ratio A/P appears because it is exactly the ratio of driving force to resisting surface.
That derivation also tells you when the ratio stops being a good summary. It assumes the shear stress is reasonably uniform around the boundary, and in very wide shallow channels or very narrow deep ones it is not. In a wide channel the hydraulic radius approaches the depth, which is fine; but in a compound channel with a deep main section and shallow flood berms, a single hydraulic radius averages together two flows with very different velocities and gives a discharge that can be badly wrong. The usual answer is to divide the section into subsections and treat each separately.
For closed non-circular ducts the hydraulic diameter is likewise an approximation rather than an identity. It works well for turbulent flow in shapes that are not too far from round, and it is noticeably less reliable for laminar flow and for extreme aspect ratios, where exact solutions differ from the hydraulic-diameter estimate by tens of per cent. The Froude number calculator covers the free-surface regime question, and the flow rate calculator and pipe volume calculator handle the volumetric arithmetic downstream.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Including the free surface in the wetted perimeter — there is no wall at the surface and no shear there, so it does not count. Including it understates R and therefore the capacity.
- Using R where the hydraulic diameter belongs — Reynolds numbers and friction factors take 4R. Using R alone understates the Reynolds number by a factor of four.
- Using the channel depth instead of the flow depth — R is a property of the water in the section, not of the excavation. A half-empty channel has a quite different R.
- Assuming a full pipe has the greatest R — it does not. R peaks at about 81 per cent depth, and a full pipe has the same R as a half-full one.
- Using one R for a compound section — a deep main channel with shallow flood berms carries flows at very different velocities, and a single averaged R can be badly misleading.
Related Free Tools From Arb Digital
To turn a hydraulic radius into a discharge, use the open channel flow calculator, which applies Manning's equation for a chosen shape and slope. The Froude number calculator tells you whether the flow is subcritical or supercritical, and the hydraulic jump calculator handles the transition between the two. For closed conduits use the pipe flow calculator and the Reynolds number calculator, both of which take the hydraulic diameter this page produces, and the pipe volume calculator or flow rate calculator for volumes and rates. For groundwater rather than surface water, see the hydraulic conductivity calculator and the hydraulic gradient calculator. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It is the cross-sectional area of the flow divided by the wetted perimeter, the length of boundary actually in contact with the fluid. It has units of length and appears in Manning's equation and in the expression for average boundary shear stress in a channel.
Because the area is pi D squared over four and the wetted perimeter is pi D, so the ratio is D over four. It is not a radius in any geometric sense, which is why it comes out as half the actual radius of the pipe.
Hydraulic diameter is four times the hydraulic radius. The factor four is chosen so that for a round pipe the hydraulic diameter equals the real diameter, which lets standard pipe-flow correlations and friction charts be used unchanged for non-circular ducts.
No. Only the solid boundary in contact with the fluid counts, because that is where shear stress acts. Including the water surface would add length that exerts no resistance and would understate the hydraulic radius and the channel's capacity.
At roughly 81 per cent of the diameter, where it reaches about 0.30 D. Above that the wetted perimeter grows faster than the added area, so the ratio falls back to 0.25 D when the pipe is completely full.
Yes. Both come to a quarter of the diameter, because halving the depth halves both the flow area and the wetted perimeter, leaving the ratio unchanged. It is a genuine result and one of the more surprising facts in open-channel hydraulics.
It works well for turbulent flow in shapes reasonably close to round. It is much less reliable for laminar flow and for extreme aspect ratios, where exact solutions can differ from the hydraulic-diameter estimate by tens of per cent.
Because Manning's equation is an empirical fit to observed channel flow, and the two-thirds exponent on hydraulic radius is what the data supported. This page reports that power directly so it can be carried into the equation without a separate step.
This tool is provided for educational and study use. It computes cross-sectional geometry for idealised prismatic sections and does not account for compound sections, non-uniform roughness, sediment deposition, vegetation or the limits of the hydraulic-diameter approximation in laminar or extreme-aspect-ratio ducts, so treat its output as a physics result rather than a design-grade figure.
