Hydraulic Radius Calculator - R = A / P across five shapes

A hydraulic radius calculator turns the geometry of a channel or pipe into the single ratio R_h = A / P that controls open-channel and pipe-flow calculations, where A is the cross-sectional flow area and P is the wetted perimeter in contact with the water. The hydraulic radius shows up in Manning's equation for gravity-driven open-channel flow, in the Darcy-Weisbach equation for closed-pipe flow, and in almost every textbook chapter on channel design.

• • Open-channel design check: Verify the hydraulic radius of a rectangular, trapezoidal, or triangular channel before plugging it into Manning's equation for flow velocity.
• • Culvert and sewer capacity review: Compute R_h for a full or partially filled circular pipe to size a storm sewer or sanitary sewer under a design discharge.
• • Irrigation ditch geometry: Pick the side slope z and bottom width b of an earthen ditch so the hydraulic radius matches a target discharge.
• • Classroom and homework problems: Work through the area-over-perimeter ratio for each canonical shape and confirm against a textbook formula.

Hydraulic radius has units of length, usually metres. A wider, deeper section has a larger hydraulic radius than a narrow, shallow section of equal area, because less of the perimeter is in contact with the water.

Once the hydraulic radius is in hand, the natural next step in a gravity-driven design is the slope-driven flow term used in the Hydraulic Gradient Calculator.

The calculator reads the selected shape and the matching geometry inputs, then computes the area A and the wetted perimeter P using the per-shape formula set before dividing A by P.

• b: Bottom width of the rectangular or trapezoidal channel in metres.
• y: Water depth in the rectangular, trapezoidal, or triangular channel in metres.
• z: Side slope of the trapezoidal or triangular channel, as horizontal run per 1 unit of vertical rise.
• r: Internal radius of the circular pipe in metres.
• h: Water depth inside a partially filled circular pipe, measured from the pipe invert to the free surface.
• theta: Central angle subtended by the water surface in the partially filled pipe, equal to 2 * arccos((r - h) / r).

According to the FHWA Hydraulic Design Series 4, Introduction to Highway Hydraulics, the hydraulic radius R is the cross-sectional flow area A divided by the wetted perimeter P (R = A / P), and for rectangular and trapezoidal channels the equivalent closed-form expressions are A = By + Zy^2, P = B + 2y sqrt(1 + Z^2), and R = (By + Zy^2) / (B + 2y sqrt(1 + Z^2)) as listed in Equations 4.8 through 4.10.

Once R_h is known, the friction factor for a full pipe can be looked up from the same geometry through the Reynolds number plus the Friction Factor Calculator.

Four ideas explain what the hydraulic radius actually measures and why it shows up in so many open-channel and pipe-flow equations.

When the open channel narrows to a critical-flow control such as a weir crest, the Broad Crested Weir Calculator uses the same A over P idea to set the upstream head for a given discharge.

The form opens with the shape selector, then shows the geometric inputs that match the chosen shape; the result panel reads the hydraulic radius, area, wetted perimeter, hydraulic diameter, and (for the partial-pipe case) the central angle.

1. 1 Pick the shape that matches your channel or pipe: Use rectangular, trapezoidal, or triangular for an open channel, full circular pipe for a pressurised pipe, and partially filled circular pipe for a surcharged sewer.
2. 2 Enter the geometric inputs for the chosen shape: Width b and depth y for the rectangle, plus side slope z for the trapezoid; pipe radius r for the full pipe; r plus flow depth h for the partial pipe.
3. 3 Read the hydraulic radius first: The primary output is R_h in metres. Compare it against the depth or radius to spot whether the cross-section is roughly square or stretched.
4. 4 Check the area and wetted perimeter: A is the cross-sectional flow area and P is the wetted perimeter; both should match the formula for the chosen shape.
5. 5 Use the hydraulic diameter for pipe-flow checks: D_h = 4 * R_h is the equivalent diameter for the Darcy-Weisbach equation; for a full pipe D_h equals the pipe diameter 2r.
6. 6 Feed the result into Manning's or Darcy-Weisbach: For open channels pass R_h into V = (1 / n) * R_h^(2/3) * S^(1/2); for pressurised pipes pass D_h into the head-loss equation.

Set shape to 'Trapezoidal channel', enter b = 2 m, y = 1 m, z = 1. The form returns A = 3 m^2, P = 4.8284 m, R_h = 0.6213 m. Pass R_h into Manning's equation with n = 0.03 and S = 0.001 to estimate V = 0.83 m/s.

When the resulting D_h is paired with a design velocity and a fluid viscosity, the Reynolds Number & Flow Regime Calculator decides whether the pipe flow is laminar or turbulent.

Five reasons a hydraulic radius calculator belongs next to open-channel and pipe-flow design spreadsheets.

• • Five shape modes in one form: Covers rectangular, trapezoidal, triangular, full pipe, and partially filled pipe without switching tools.
• • R = A / P applied per shape: Returns the per-shape hydraulic radius formula and exposes both A and P so intermediate values can be sanity-checked.
• • Hydraulic diameter included for pipe-flow work: Returns D_h = 4 * R_h so the result feeds directly into the Darcy-Weisbach equation.
• • Partial-pipe arc geometry made visible: Surfaces the central angle theta in radians so the user can confirm the arc geometry behind the segment area.
• • Useful for coursework, drainage review, and irrigation design: Lets a student check a textbook problem, a drainage engineer size a culvert, and an irrigation designer compare ditch cross-sections.

When the open channel ends at a spillway or chute and the flow transitions to supercritical, the Hydraulic Jump Calculator builds on the same rectangular-channel vocabulary to size the conjugate depth and jump length.

Five factors decide whether the hydraulic radius from this calculator is a design number or a screening value.

• • The calculator assumes a prismatic channel with constant shape along the flow direction and a horizontal bed. A real channel with varying cross-section, sloping bed, or sediment build-up needs a numerical backwater model.
• • The partial-pipe arc geometry uses the central angle from a horizontal water surface. Tilted pipes, dipping inverts, or air pockets at the crown break that assumption.

According to the FHWA HEC-18, Evaluating Scour at Bridges, hydraulic radius is the cross-sectional area of a stream divided by its wetted perimeter, written R = A / P, and the same ratio enters the average bed-shear stress as tau_0 = gamma * R * S, with the worked contraction-scour example on page 8-8 returning R = 320 m^2 / 122 m = 2.62 m and tau_0 = 9810 N/m^3 * 2.62 m * 0.002 m/m = 51.4 Pa for the approach channel.

When the hydraulic radius feeds into a closed-pipe energy balance, the Bernoulli Equation Calculator carries the same D_h over to the head-loss term in the Bernoulli equation.