Hydraulic Radius Calculator

The hydraulic radius (R) is the ratio of the cross-sectional flow area (A) to the wetted perimeter (P) of a channel or pipe: R = A / P. It is a key parameter in open-channel hydraulics and pipe flow used to characterize how efficiently a channel conveys flow relative to the friction from its boundary.

For a fully flowing circular pipe of radius r, the entire inner circumference is in contact with the fluid, so the wetted perimeter is P = 2πr. The flow area equals πr², giving a hydraulic radius of R = πr² / (2πr) = r/2.

For a pipe partially filled to depth h with radius r, you first compute the central angle θ = 2 × arccos[(r − h) / r]. The wetted perimeter is P = r × θ and the flow area is A = (r² / 2) × (θ − sin θ). The hydraulic radius is then R = A / P.

For a rectangular channel of bottom width b and flow depth y, the flow area is A = b × y and the wetted perimeter is P = b + 2y. Therefore, the hydraulic radius is R = (b × y) / (b + 2y).

For a trapezoidal channel with bottom width b, flow depth y, and side slope z (horizontal per unit vertical), the flow area is A = (b + z × y) × y and the wetted perimeter is P = b + 2y × √(1 + z²). The hydraulic radius is R = A / P.

For a triangular channel with side slope z and flow depth y, the flow area is A = z × y² and the wetted perimeter is P = 2y × √(1 + z²). The hydraulic radius is R = (z × y) / (2 × √(1 + z²)).

The hydraulic radius appears in key equations like the Manning equation and the Darcy-Weisbach equation, which are used to predict flow velocity and discharge in pipes and open channels. A larger hydraulic radius generally indicates more efficient flow with less frictional resistance per unit of flow area.

No. Hydraulic diameter (Dh) is defined as four times the hydraulic radius: Dh = 4R = 4A/P. For a full circular pipe, R = r/2 and Dh = 2r = D (the pipe diameter), which is the intuitive result. Hydraulic diameter is commonly used in the Darcy-Weisbach equation for pipe flow.