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Walk-through

How to Use This Calculator

3 steps

1

Pick a channel shape and enter its dimensions

On the Hydraulic Radius tab, choose Rectangular, Circular (full pipe), or Trapezoidal from the shape selector. Each shape shows its own fields — bottom width and depth for rectangular and trapezoidal channels, diameter for a full pipe, plus a side slope for trapezoids. The calculator updates instantly as you type.

2

Read the hydraulic radius, area, and wetted perimeter

The result card shows Rh in meters, along with the flow area A and wetted perimeter P used to compute it (Rh = A/P). For a full pipe, Rh always reduces to exactly D/4 — a useful sanity check.

3

Compare shapes or work backward from A and P

The By Shape tab compares Rh across all three shapes for the same dimensions, so you can see how geometry alone changes it. The Area & Perimeter tab skips the shape math entirely — enter an already-known area and wetted perimeter to get Rh = A/P directly.

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Reference

Formula & Methodology

4 formulas

Hydraulic radius

Rh = A / P

Flow area A (m²) divided by wetted perimeter P (m) — the length of channel boundary actually in contact with the fluid. Rh has units of length (m) and is the key geometric term in open-channel flow equations like Manning's and Chezy's.

Rectangular open channel

A = b·y, P = b + 2y

Bottom width b times flow depth y gives the area. The wetted perimeter is the bottom plus both sides — the top is open to air and not wetted.

Full circular pipe

A = πD²/4, P = πD, Rh = D/4

When a circular pipe runs completely full, the entire circumference is wetted. The area-over-perimeter ratio simplifies exactly to one quarter of the diameter, independent of any other calculation.

Trapezoidal channel

A = y(b + zy), P = b + 2y√(1+z²)

Bottom width b, depth y, and side slope z (horizontal run per unit of vertical rise, expressed as z:1). Setting z = 0 collapses the trapezoid formulas to the rectangular case exactly.

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Glossary

Key Terms Explained

6 terms

Hydraulic radius The ratio of a channel's cross-sectional flow area to its wetted perimeter (Rh = A/P), measured in meters. It's not a true geometric radius — the name is historical — but a shape factor that captures how efficiently a cross-section carries flow relative to its friction-causing boundary.

Wetted perimeter The length of the channel or pipe boundary that is in direct contact with the flowing fluid. For an open channel this excludes the free water surface at the top; for a full pipe it's the entire circumference.

Flow area The cross-sectional area of the fluid perpendicular to the direction of flow — the area a slice through the channel or pipe would expose.

Open channel A conduit, such as a canal, ditch, or partially-full pipe, where the flow has a free surface exposed to the atmosphere rather than being fully enclosed and pressurized.

Manning equation A widely used empirical formula for open-channel flow velocity, v = (1/n)·Rh^(2/3)·S^(1/2), where n is a roughness coefficient and S is the channel slope. Hydraulic radius is the geometric term that feeds directly into it.

Side slope For a trapezoidal channel, the ratio of horizontal run to vertical rise of the sloped banks, written z:1. A larger z means a gentler, more spread-out bank; z = 0 gives vertical walls (a rectangular channel).

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Scenarios

Real-World Examples

3 worked examples

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Irrigation ditch

A rectangular open channel 2 m wide flowing 0.5 m deep

Shape Rectangular Bottom width b 2 m Flow depth y 0.5 m

A = 2 × 0.5 = 1 m², P = 2 + 2(0.5) = 3 m, so Rh = 1/3 ≈ 0.333 m. This value would plug directly into Manning's equation to estimate flow velocity for the ditch.

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Full storm drain pipe

A 0.3 m diameter pipe running completely full

Shape Circular (full) Diameter D 0.3 m

For a full pipe, Rh always equals D/4 regardless of the diameter — here 0.3/4 = 0.075 m. This shortcut avoids computing area and circumference separately whenever the pipe is known to be flowing completely full.

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Sloped-bank drainage channel

A trapezoidal channel with a 1 m bottom, 0.6 m depth, and 1.5:1 side slopes

Shape Trapezoidal Bottom width b 1 m Flow depth y 0.6 m Side slope z 1.5

A = 0.6 × (1 + 1.5 × 0.6) = 1.14 m², P = 1 + 2 × 0.6 × √(1 + 1.5²) ≈ 3.16 m, so Rh ≈ 0.361 m. Sloped banks add wetted perimeter without adding as much area, which is why trapezoidal channels sit between rectangular and triangular shapes in hydraulic efficiency.

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Deep Dive

What Hydraulic Radius Measures and Why It Matters for Channel Flow

3 chapters · 4 min read

Hydraulic radius is the single geometric number that open-channel and pipe-flow formulas — most famously Manning's equation — use to capture how a cross-section's shape affects friction and flow capacity. This calculator computes Rh = A/P for the three most common channel shapes, lets you compare shapes side by side, and can work backward from a known area and wetted perimeter.

How the Hydraulic Radius Calculator works

Hydraulic radius is a ratio: flow area A divided by wetted perimeter P. A bigger Rh means a cross-section carries more flow area for each unit of boundary that generates friction — which is why a full circular pipe (Rh = D/4) and a wide, shallow rectangular channel behave very differently even at the same flow area. This calculator applies the correct A and P formulas for rectangular, full-circular, and trapezoidal cross-sections, then divides to get Rh. For a full pipe, no other calculation is even needed — Rh reduces algebraically to exactly one quarter of the diameter.

Inputs and what they mean

Bottom width (b) and flow depth (y) describe the channel's footprint at the waterline; diameter (D) is used only for the full-pipe case. Side slope (z) applies to trapezoidal channels and expresses how far the bank runs horizontally for every unit it rises vertically — a z of 1.5 means the bank runs 1.5 m sideways for every 1 m of depth. Setting z to 0 turns a trapezoidal channel into a rectangular one with identical results, which is a useful way to sanity-check the trapezoid formula against the simpler rectangular one.

Limits and edge cases

This calculator assumes a partially or completely full, prismatic (constant cross-section) channel or pipe with no irregular bottom, vegetation, or obstructions — real wetted perimeters can be higher when the channel bed is rough or irregular. The full-pipe circular case only applies when the pipe is running 100% full; a partially full circular pipe has a different, more involved area-and-perimeter formula not covered here. Hydraulic radius itself doesn't predict flow velocity or capacity on its own — it's an input to Manning's or Chezy's equation, which also needs a roughness coefficient and channel slope.

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Questions

Frequently Asked Questions

6 questions

What is the formula for hydraulic radius?+

Rh = A/P, where A is the cross-sectional flow area (m²) and P is the wetted perimeter (m) — the length of channel boundary actually in contact with the fluid. Rh has units of length (meters).

What is the hydraulic radius of a full pipe?+

For any circular pipe flowing completely full, Rh always equals D/4 — one quarter of the diameter. This comes directly from dividing the full-circle area (πD²/4) by the full circumference (πD).

What is hydraulic radius used for?+

It's the key geometric term in open-channel flow formulas, most notably Manning's equation (v = (1/n)·Rh^(2/3)·S^(1/2)), which estimates flow velocity from a channel's roughness, slope, and cross-sectional shape.

What does wetted perimeter mean?+

The length of the channel or pipe boundary that's actually touching the flowing fluid. In an open channel this excludes the free surface at the top — only the bottom and sides count — while in a full pipe the entire circumference is wetted.

What units does the hydraulic radius calculator use?+

All dimensions are entered in meters (m), flow area is reported in square meters (m²), and hydraulic radius is reported in meters (m), matching the standard SI convention used in Manning's equation.

Does this calculator support trapezoidal channels?+

Yes. Enter the bottom width, flow depth, and side slope (z, expressed as horizontal run per unit of vertical rise) on the Hydraulic Radius tab, or compare it against rectangular and circular shapes on the By Shape tab.

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Hydraulic Radius Calculator

Channel shape and dimensions