{
  "id": "manning-equation",
  "version": "9167e7631b67",
  "status": "published",
  "name": "Manning Equation Calculator",
  "question": "Manning equation: what is the flow?",
  "summary": "Manning equation calculator: velocity and discharge of uniform flow in a rectangular, trapezoidal, triangular or circular channel from its size, flow depth, slope and Manning roughness n.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/manning-equation-calculator",
  "markdown": "https://www.acalculator.org/physics/manning-equation-calculator.md",
  "kind": "function",
  "method": "V = (1 ÷ n) × R^(2/3) × S^(1/2) in metres (1.486 ÷ n in feet), Q = V × A, R = A ÷ P; rectangle A = by, P = b + 2y; trapezoid A = (b + zy)y, P = b + 2y√(1 + z²); triangle A = zy², P = 2y√(1 + z²); pipe θ = 2 arccos(1 − 2y ÷ D), A = D²(θ − sin θ) ÷ 8, P = Dθ ÷ 2.",
  "assumptions": [
    "Steady, uniform flow: the depth and velocity stay the same along the channel, and the water surface is parallel to the bed.",
    "n is the normal value for the lining; real channels vary between the table’s minimum and maximum."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "sh": {
        "title": "Channel shape",
        "description": "The shape of the channel or pipe cross-section.",
        "type": "string",
        "enum": [
          "trap",
          "rect",
          "tri",
          "circle"
        ]
      },
      "b": {
        "title": "Bottom width (b)",
        "description": "The width of the channel bed.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.0001,
        "maximum": 10000
      },
      "z": {
        "title": "Side slope (z:1)",
        "description": "Horizontal run of each side wall per unit of height, for example 2 for 2:1.",
        "type": "number",
        "minimum": 0,
        "maximum": 100
      },
      "D": {
        "title": "Pipe diameter (D)",
        "description": "The inside diameter of the pipe.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.0001,
        "maximum": 10000
      },
      "y": {
        "title": "Flow depth (y)",
        "description": "The depth of water at the deepest point.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.0001,
        "maximum": 10000
      },
      "S": {
        "title": "Slope (S, ft/ft or m/m)",
        "description": "The fall of the channel bed per unit of length, for example 0.001 for 1 in 1,000 (0.1%).",
        "type": "number",
        "minimum": 1e-7,
        "maximum": 1
      },
      "mat": {
        "title": "Lining",
        "description": "The channel or pipe material, which sets Manning’s n (USACE HEC-RAS, normal values).",
        "type": "string",
        "enum": [
          "concretePipe",
          "concretePipeBends",
          "castIron",
          "steel",
          "cmp",
          "lucite",
          "trowel",
          "float",
          "unfinished",
          "gunite",
          "asphalt",
          "brick",
          "earthNew",
          "earth",
          "gravel",
          "grass",
          "stream",
          "winding",
          "mountain",
          "weedy",
          "custom"
        ]
      },
      "n": {
        "title": "Manning’s n",
        "description": "The roughness coefficient of the channel.",
        "type": "number",
        "minimum": 0.005,
        "maximum": 0.5
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Discharge",
      "description": "Q = V × A, in cubic feet and cubic metres per second.",
      "format": "text"
    },
    "cfs": {
      "label": "Discharge (ft³/s)",
      "description": "The flow rate in cubic feet per second.",
      "format": "number"
    },
    "cms": {
      "label": "Discharge (m³/s)",
      "description": "The flow rate in cubic metres per second.",
      "format": "number"
    },
    "lps": {
      "label": "Discharge (L/s)",
      "description": "The flow rate in litres per second.",
      "format": "number"
    },
    "gpm": {
      "label": "Discharge (gal/min)",
      "description": "The flow rate in US gallons per minute.",
      "format": "number"
    },
    "fps": {
      "label": "Velocity (ft/s)",
      "description": "The mean velocity V = (k ÷ n) × R^(2/3) × S^(1/2), in feet per second.",
      "format": "number"
    },
    "mps": {
      "label": "Velocity (m/s)",
      "description": "The mean velocity in metres per second.",
      "format": "number"
    },
    "area": {
      "label": "Flow area (A)",
      "description": "The area of the water in the cross-section.",
      "format": "quantity",
      "unit": "ft2"
    },
    "perimeter": {
      "label": "Wetted perimeter (P)",
      "description": "The length of channel wall the water touches.",
      "format": "quantity",
      "unit": "ft"
    },
    "radius": {
      "label": "Hydraulic radius (R)",
      "description": "R = A ÷ P.",
      "format": "quantity",
      "unit": "ft"
    },
    "top": {
      "label": "Top width (T)",
      "description": "The width of the water surface.",
      "format": "quantity",
      "unit": "ft"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "sh": "trap",
      "b": 3.048,
      "z": 2,
      "D": 0.9144000000000001,
      "y": 0.9144000000000001,
      "S": 0.001,
      "mat": "earth",
      "n": 0.022
    },
    "outputs": {
      "result": "Q = 165.44 ft³/s (4.6846 m³/s)",
      "cfs": 165.4350566844598,
      "cms": 4.684599121072672,
      "lps": 4684.5991210726725,
      "gpm": 74252.40985733677,
      "fps": 3.446563680926245,
      "mps": 1.0505126099463196,
      "area": 4.4593459200000005,
      "perimeter": 7.137321117251616,
      "radius": 0.6247926703509692,
      "top": 6.7056000000000004
    },
    "text": "Manning’s equation gives Q = 165.44 ft³/s (4.6846 m³/s) at a mean velocity of 3.4466 ft/s."
  },
  "examples": [
    {
      "given": {
        "sh": "rect",
        "b": 2,
        "y": 1,
        "S": 0.001,
        "mat": "trowel"
      },
      "expect": {
        "mps": 1.5323923806378645,
        "cms": 3.064784761275729,
        "radius": 0.5
      },
      "source": "USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); USACE HEC-RAS Hydraulic Reference Manual, Table 3-1 Manning’s n Values (Chow, 1959), https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/basic-data-requirements/geometric-data/energy-loss-coefficients (retrieved 2026-10-18) (trowel-finished concrete, 0.013); hand calculation in content.mdx: R = 2 ÷ 4 = 0.5 m, V = 0.5^(2/3) × √0.001 ÷ 0.013",
      "tolerance": 1e-12
    },
    {
      "given": {
        "sh": "trap",
        "b": 3.048,
        "z": 2,
        "y": 0.9144000000000001,
        "S": 0.001,
        "mat": "earth"
      },
      "expect": {
        "cfs": 165.4350566844598,
        "fps": 3.446563680926245,
        "area": 4.45934592
      },
      "source": "USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); USACE HEC-RAS Hydraulic Reference Manual, Table 3-1 Manning’s n Values (Chow, 1959), https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/basic-data-requirements/geometric-data/energy-loss-coefficients (retrieved 2026-10-18) (earth, clean, after weathering, 0.022); hand calculation in content.mdx: A = (10 + 2 × 3) × 3 = 48 ft², P = 10 + 6√5 = 23.416 ft",
      "tolerance": 1e-12
    },
    {
      "given": {
        "sh": "circle",
        "D": 1,
        "y": 1,
        "S": 0.005,
        "mat": "concretePipeBends"
      },
      "expect": {
        "mps": 2.1585808621334097,
        "cms": 1.6953454446644605,
        "radius": 0.25,
        "top": 0
      },
      "source": "USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); USACE HEC-RAS Hydraulic Reference Manual, Table 6-1 Manning’s n for Closed Conduits Flowing Partly Full (Chow, 1959), https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/modeling-culverts/culvert-data-and-coefficients/manning-s-roughness-coefficient (retrieved 2026-10-18) (concrete culvert with bends, 0.013); hand calculation in content.mdx: A = π ÷ 4, P = π, R = 0.25 m",
      "tolerance": 1e-12
    },
    {
      "given": {
        "sh": "circle",
        "D": 1,
        "y": 0.5,
        "S": 0.005,
        "mat": "concretePipeBends"
      },
      "expect": {
        "mps": 2.1585808621334097,
        "cms": 0.8476727223322302,
        "top": 1
      },
      "source": "USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); hand calculation in content.mdx: θ = 2 × arccos(0) = π, A = π ÷ 8, P = π ÷ 2, R = 0.25 m",
      "tolerance": 1e-12
    },
    {
      "given": {
        "sh": "tri",
        "z": 3,
        "y": 0.5,
        "S": 0.01,
        "mat": "grass"
      },
      "expect": {
        "cms": 1.0643186561021574,
        "area": 0.75
      },
      "source": "USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); USACE HEC-RAS Hydraulic Reference Manual, Table 3-1 Manning’s n Values (Chow, 1959), https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/basic-data-requirements/geometric-data/energy-loss-coefficients (retrieved 2026-10-18) (short grass, 0.027); hand calculation in content.mdx: A = 3 × 0.5² = 0.75 m², P = 2 × 0.5 × √10",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "USACE, HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model, Basic Concepts and Equations: Q = (K ÷ N) × R^(2/3) × A × √S with K = 1.486 in English units and 1.0 in metric units; Solution of Equations: trapezoid wetted perimeter W + 2Y√(1 + Z²), retrieved 2026-10-18. https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model",
    "USACE, HEC-RAS Hydraulic Reference Manual, Energy Loss Coefficients: Table 3-1, Manning’s n values for channels (from Chow, Open-Channel Hydraulics, 1959), retrieved 2026-10-18. https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/basic-data-requirements/geometric-data/energy-loss-coefficients",
    "USACE, HEC-RAS Hydraulic Reference Manual, Manning’s Roughness Coefficient: Table 6-1, Manning’s n for closed conduits flowing partly full (Chow, 1959), retrieved 2026-10-18. https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/modeling-culverts/culvert-data-and-coefficients/manning-s-roughness-coefficient",
    "USGS, Surface-Water Field Techniques, Verified Roughness Characteristics of Natural Channels (from Water-Supply Paper 1849, H. H. Barnes Jr.): photographs and measured n of 22 natural channels, 0.024 to 0.079, retrieved 2026-10-19. https://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm",
    "Wolfram MathWorld, Circular Segment: segment area ½R²(θ − sin θ) and arc length Rθ, retrieved 2026-10-18. https://mathworld.wolfram.com/CircularSegment.html",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 ft³ = 0.028316846592 m³, 1 US gallon = 3.785411784 L, retrieved 2026-10-18. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "flow-rate",
    "reynolds-number",
    "pipe-volume"
  ],
  "changelog": []
}