Free Hydraulic Radius Calculator

Understanding Hydraulic Radius in Open Channel Flow

Hydraulic radius is a vital parameter in open channel hydraulics, used to characterize the efficiency of a channel cross-section in conveying water. It appears in the Manning equation and other resistance formulas, linking flow geometry to velocity and discharge. This online hydraulic radius calculator simplifies the process by automatically applying the correct hydraulic radius equation for rectangular, trapezoidal, triangular, and circular pipe cross-sections — whether the pipe flows fully or partially full. In addition, the tool can function as a wetted perimeter calculator and a channel flow calculator, providing the essential geometric measures needed for comprehensive flow analysis.

The Basic Definition

The hydraulic radius $R$ is defined as:

$$R = \frac{A}{P}$$

where:

• $A$ = cross‑sectional area of the flow,
• $P$ = wetted perimeter (the length of the channel boundary in contact with the fluid).

Obtaining $A$ and $P$ for a given channel shape is the first step in determining its hydraulic radius.

Rectangular Channel

For a rectangular channel of width $b$ and flow depth $y$:

$$A = b \cdot y, \qquad P = b + 2y$$

Thus:

$$R = \frac{b \cdot y}{b + 2y}$$

This simple expression is often the starting point for many open‑channel problems.

Trapezoidal Channel

A trapezoidal channel has a bottom width $b$, depth $y$, and side slopes defined by a slope factor $z$, which represents the horizontal distance gained per unit vertical rise. The top width becomes $B = b + 2zy$.

The flow area combines the central rectangle and the two triangular side portions:

$$A = b y + z y^{2}$$

The wetted perimeter includes the bottom and the two sloping sides:

$$P = b + 2 y \sqrt{1 + z^{2}}$$

Therefore:

$$R = \frac{b y + z y^{2}}{b + 2 y \sqrt{1 + z^{2}}}$$

Trapezoidal channels are widely used in irrigation canals and roadside ditches.

Triangular Channel

For a triangular channel with side slope $z$ and depth $y$:

$$A = y^{2} z$$ $$P = 2 y \sqrt{1 + z^{2}}$$

Hence:

$$R = \frac{y^{2} z}{2 y \sqrt{1 + z^{2}}} = \frac{y z}{2 \sqrt{1 + z^{2}}}$$

Triangular channels are common in small drainage swales and laboratory flumes.

Circular Pipe

The hydraulic radius of a circular pipe depends on whether it flows full or partially full.

Full Pipe

When the pipe is completely full:

$$A = \pi r^{2}, \quad P = 2 \pi r$$ $$R = \frac{\pi r^{2}}{2 \pi r} = \frac{r}{2}$$

Thus, for a full pipe the hydraulic radius is simply half the pipe’s radius (or a quarter of the diameter).

Partially Full Pipe

For a pipe with radius $r$ and flow depth $h$ ($0 \le h \le 2r$), the water surface subtends a central angle $\theta$ given by:

$$\theta = 2 \arccos\!\left(\frac{r - h}{r}\right)$$

The wetted perimeter and flow area are:

$$P = r \theta$$ $$A = \frac{r^{2}}{2} (\theta - \sin \theta)$$

Combining them yields the hydraulic radius for a partially filled pipe:

$$R = \frac{r^{2} (\theta - \sin \theta)}{2 r \theta} = \frac{r (\theta - \sin \theta)}{2 \theta}$$

This formula is essential for sewer design and partially filled conduit analysis.

By entering the required dimensions into this hydraulic radius calculator (also usable as a wetted perimeter calculator and cross sectional area of flow tool), you can instantly obtain $R$ for any of the described shapes. The geometric outputs feed directly into further evaluations such as Manning’s equation or velocity‑discharge calculations, streamlining your open channel hydraulics work.