Formula and Method for Hydraulic Radius
Hydraulic radius (R) describes how efficiently a channel or pipe cross-section carries flow. It is defined as the cross-sectional area of the flow (A) divided by the wetted perimeter (P) — the portion of the channel or pipe boundary that is actually touching the liquid: R = A / P. A larger hydraulic radius means less wall friction per unit of flow area, so water moves more efficiently — this is why engineers use R (via Manning's or Chézy's equation) to estimate flow velocity and capacity in canals, culverts, and storm drains.
How the calculation works
Pick the shape that matches your channel or pipe, enter its dimensions, and the calculator derives A and P from standard geometric relations, then divides to get R:
- Rectangular channel (bottom width b, flow depth y): A = b·y, P = b + 2y
- Trapezoidal channel (bottom width b, depth y, side slope z horizontal per 1 vertical): A = y·(b + z·y), P = b + 2y√(1 + z²)
- Triangular channel (depth y, side slope z): A = z·y², P = 2y√(1 + z²)
- Circular pipe, flowing full (diameter D): A = πD²/4, P = πD, which simplifies to R = D/4
- Circular pipe, flowing partially full (diameter D, depth y): using the central angle θ = 2·cos⁻¹(1 − 2y/D) in radians, A = (D²/8)·(θ − sin θ) and P = (D/2)·θ
In every case the free water surface — the open top of a channel, or the dry arc above the waterline in a partially full pipe — is excluded from the wetted perimeter, since it is not in contact with a solid boundary. The calculator also reports the hydraulic diameter, D_h = 4R, which is the value to plug into circular-pipe-based correlations (Reynolds number, Darcy-Weisbach friction factor) when the actual cross-section is not circular.
Common mistakes
- Treating hydraulic radius as a literal radius: for a full circular pipe, R = D/4, not D/2 — it is a flow-efficiency ratio, not a distance from the pipe's center.
- Including the free surface in the wetted perimeter: only the solid bed and walls (or pipe wall) count; the open top of a channel is excluded.
- Mixing units: keep width, depth, and diameter in the same length unit before calculating — the tool assumes all length inputs share the selected unit.
- Confusing hydraulic radius with hydraulic diameter: R and D_h = 4R are related but not interchangeable; formulas written for D_h (like Reynolds number) will be off by a factor of 4 if you substitute R directly.
Real-world applications
- Open-channel design (irrigation canals, drainage ditches, natural streams) uses R in Manning's equation, V = (1/n)·R^(2/3)·S^(1/2), to estimate flow velocity and channel capacity.
- Storm sewer and culvert engineers calculate R for pipes flowing partially full, since sanitary and storm pipes are rarely designed to run completely full.
- HVAC duct design uses hydraulic diameter (4R) to apply circular-duct friction-loss charts to rectangular or irregular ductwork.
- Hydraulics and fluid mechanics use R to compute the Reynolds number for non-circular conduits, determining whether flow is laminar or turbulent.