Hydrology · Stormwater Drainage

Rational Method (Q = CiA): Formula, Runoff Coefficient Table, Time of Concentration & Worked Examples

The rational method is the most widely used way to size storm drains, culverts and gutters for small catchments. This guide covers the formula in SI and US units, where each term comes from, how to pick C, how to compute the time of concentration and read an IDF curve, two fully worked examples, the mistakes that most often produce wrong designs, and an interactive calculator that shows why the critical storm duration equals t_c.

SI & US formulasC table4 t_c methodsInteractive calculator

By Bimal Ghimire · Published October 28, 2026 · 18 min read

Q = CiA

Peak runoff Q equals the runoff coefficient C times the rainfall intensity i (for a storm lasting the time of concentration) times the drainage area A. In SI: $Q\,[\text{m}^3/\text{s}] = C\,i\,A/360$ with i in mm/h and A in ha. In US units: $Q\,[\text{cfs}] = C\,i\,A$ with i in in/h and A in acres. Use it for catchments up to roughly 80 ha (200 ac).

The Equation

Rational Method Formula in SI and US Units

The rational method dates to Mulvaney (1851) in Ireland and Kuichling (1889) in the US. Its logic is a steady-state water balance: if rain falls at a constant intensity $i$ on an area $A$ for long enough, the outflow rate approaches the inflow rate $iA$, reduced by the fraction lost to infiltration and storage, which is $1-C$.

General form

$$Q = k\,C\,i\,A$$

System	Q	i	A	Conversion k
US customary	cfs (ft³/s)	in/h	acres	1.008 ≈ 1
SI (hectares)	m³/s	mm/h	ha	1/360 = 0.00278
SI (km²)	m³/s	mm/h	km²	1/3.6 = 0.278
SI (litres)	L/s	mm/h	ha	2.78

Where 360 comes from: 1 mm/h on 1 ha = 0.001 m × 10,000 m² per 3,600 s = 1/360 m³/s. In US units, 1 in/h on 1 acre = (1/12 ft)(43,560 ft²)/3,600 s = 1.008 cfs, which is why the US form needs no constant.

Validity

Assumptions and Limitations

Rainfall is uniform in space and time over the whole catchment for a duration at least equal to t_c. This is only reasonable for small areas.
Peak flow happens when the whole catchment contributes, i.e. when storm duration equals t_c.
C is constant during the storm and the same for all return periods (corrected for large storms with a frequency factor).
The return period of Q equals the return period of i. A 10-year intensity gives a 10-year peak.
No significant storage (ponds, wetlands, floodplains) in the catchment.

Size limit: TxDOT's Hydraulic Design Manual restricts the method to drainage areas up to about 200 acres (80 ha) without significant flood storage. Other agencies use limits from about 20 to 200 acres; check your local manual. For larger watersheds use the SCS curve number method with a unit hydrograph.

The rational method gives only a peak, not a hydrograph or a runoff volume. It cannot size detention ponds on its own (the "modified rational method" extends it with a trapezoidal hydrograph for preliminary detention sizing).

Coefficient

Runoff Coefficient (C) Table

C is the most subjective input. It lumps infiltration, depression storage, evaporation and attenuation into one number, so use local design-manual values where they exist.

By land use (TxDOT Hydraulic Design Manual, Table 4-10)

Land use / surface	C
Downtown business areas	0.70–0.95
Single-family residential	0.30–0.50
Apartment dwelling areas	0.30–0.70
Light industrial	0.30–0.80
Heavy industrial	0.60–0.90
Parks, cemeteries	0.10–0.25
Asphaltic streets	0.85–0.95
Concrete streets	0.90–0.95
Roofs	0.75–0.95

Lawns and other surfaces (ASCE & WPCF 1969, as reproduced in Chow et al. 1988)

Surface	Flat (≤ 2%)	Average (2–7%)	Steep (≥ 7%)
Lawns, sandy soil	0.05–0.10	0.10–0.15	0.15–0.20
Lawns, heavy (clay) soil	0.13–0.17	0.18–0.22	0.25–0.35

Rural watersheds (TxDOT Table 4-11): C = C_r + C_i + C_v + C_s

Component	Extreme	High	Normal	Low
Relief C_r	0.28–0.35	0.20–0.28	0.14–0.20	0.08–0.14
Soil infiltration C_i	0.12–0.16	0.08–0.12	0.06–0.08	0.04–0.06
Vegetal cover C_v	0.12–0.16	0.08–0.12	0.06–0.08	0.04–0.06
Surface storage C_s	0.10–0.12	0.08–0.10	0.06–0.08	0.04–0.06

Composite C and frequency factor

$$C_w=\frac{\sum C_j A_j}{\sum A_j},\qquad Q = C_f\,C\,i\,A,\quad C_f\,C\le 1$$

Many design manuals use a frequency factor C_f = 1.0 for 2–10-year, 1.1 for 25-year, 1.2 for 50-year and 1.25 for 100-year storms, because large storms satisfy depression storage and infiltration faster. Some manuals (including TxDOT) do not use C_f; follow your governing manual.

Timing

Time of Concentration (t_c) Formulas

The time of concentration is the travel time of runoff from the hydraulically most distant point to the outlet. It sets the design storm duration, so errors in t_c go straight into i and Q. Empirical formulas differ a lot because they were built for different flow types.

Kirpich (1940): channel flow, small rural watersheds

$$t_c = 0.0078\,k\,L^{0.77}S^{-0.385}\ \ [\text{L in ft}],\qquad t_c = 0.0195\,k\,L^{0.77}S^{-0.385}\ \ [\text{L in m}]$$

t_c in minutes, S in m/m (ft/ft). Adjustment k: 1.0 natural channels/bare soil, 2.0 general overland flow on grass, 0.4 concrete or asphalt overland flow, 0.2 concrete channels. Developed from Tennessee watersheds of 1–112 acres with slopes 3–10%.

FAA (1970): overland flow, urban and airfields

$$t_c = \frac{1.8\,(1.1-C)\,L^{0.5}}{S^{1/3}}\ \ [\text{L ft, S \%}],\qquad t_c = \frac{3.26\,(1.1-C)\,L^{0.5}}{S^{1/3}}\ \ [\text{L m, S \%}]$$

NRCS lag method

$$t_c = \frac{60\,L^{0.8}\left(\frac{1000}{CN}-9\right)^{0.7}}{1140\,Y^{0.5}}\quad[\text{min; L ft, Y watershed slope \%}]$$

Kerby (1959): short overland flow

$$t_c = 0.8268\left(\frac{L\,r}{S^{0.5}}\right)^{0.467}\ [\text{L ft}],\qquad t_c = 1.44\left(\frac{L\,r}{S^{0.5}}\right)^{0.467}\ [\text{L m}]$$

Retardance r: 0.02 smooth pavement, 0.10 smooth bare soil, 0.30 poor grass, 0.40 average grass, 0.60 deciduous forest, 0.80 conifer forest or dense grass. Intended for flow lengths under about 365 m (1,200 ft).

Same catchment, different answers. For the worked example below (L = 650 m, S = 1.5%, C = 0.42), Kirpich gives 14.4 min while FAA gives 49.3 min, and the peak flow differs by almost a factor of two (1.93 vs 1.02 m³/s). Kirpich assumes channelised flow; FAA assumes sheet flow over the full length. Real paths are usually segmented: sheet flow (≤ 100 ft in NRCS TR-55) + shallow concentrated flow + channel flow, with travel times added. Many agencies also impose a minimum t_c (TxDOT: 10 minutes).

Rainfall

Rainfall Intensity from IDF Curves

The intensity $i$ is read from an intensity–duration–frequency (IDF) curve for the design return period, at a duration equal to t_c. IDF relationships are often fitted as:

$$i=\frac{a}{(t+b)^{n}}\qquad\text{or}\qquad i=\frac{a\,T^{m}}{(t+b)^{n}}$$

t = duration (min), T = return period (years); a, b, n, m are fitted locally.

United States: NOAA Atlas 14 through the Precipitation Frequency Data Server (PFDS). NOAA Atlas 15, which accounts for climate change, is in development; NOAA's schedule lists preliminary CONUS estimates no earlier than September 2026 and published estimates no earlier than September 2027.
Elsewhere: national meteorological or hydrology departments publish IDF curves or equations; in data-scarce regions, IDF curves are derived from annual maximum rainfall series fitted with Gumbel or GEV distributions.

Why duration = t_c? If the storm is shorter than t_c, only the fraction t/t_c of the catchment contributes; if longer, intensity drops. Under the method's assumptions the peak is $Q(t)=C\,i(t)\,A\cdot\min(1,\,t/t_c)$, which is maximal at $t=t_c$. The calculator plots this curve.

Practice

Worked Example 1 (SI): Small Urban Catchment

Given: a 12 ha catchment with 4 ha of roofs and paved surfaces (C = 0.90), 5 ha of lawns on heavy soil, flat (C = 0.20), and 3 ha of park (C = 0.15). Longest flow path L = 650 m with a fall of 9.75 m, mostly in a natural channel. Local 10-year IDF (illustrative): $i = 1600/(t+12)^{0.75}$ mm/h.

Composite C: $C = (0.90\times4 + 0.20\times5 + 0.15\times3)/12 = 5.05/12 = 0.421$
Slope: $S = 9.75/650 = 0.015$ m/m
t_c by Kirpich (k = 1): $t_c = 0.0195 \times 650^{0.77} \times 0.015^{-0.385} = 14.4$ min (above the 10-minute minimum)
Intensity: $i = 1600/(14.4+12)^{0.75} = 137.4$ mm/h
Peak flow: $Q = 0.421 \times 137.4 \times 12 / 360 = 1.93$ m³/s (≈ 1,930 L/s)

This example is pre-loaded in the calculator. Switch the t_c method to FAA to see the peak fall to about 1.02 m³/s, a reminder to choose the t_c method that matches the actual flow path.

Practice

Worked Example 2 (US): 25-Year Design with Frequency Factor

Given: A = 25 acres, composite C = 0.55, t_c = 18 min, and the 25-year IDF curve gives i = 4.2 in/h at 18 minutes. The governing manual uses C_f = 1.1 for 25-year storms.

Check C_fC ≤ 1: 1.1 × 0.55 = 0.605 ✓
Peak flow: $Q = C_f\,C\,i\,A = 0.605 \times 4.2 \times 25 = 63.5$ cfs (≈ 1.80 m³/s)

Rational Method Calculator

1 · Runoff coefficient (composite)

Return period

2 · Time of concentration

Method

Flow length L (m)

Slope (%)

Kirpich k

Curve number CN

Kerby r

t_c (min)

Minimum t_c (min)

3 · Rainfall intensity

Source

b (min)

i (mm/h)

IDF coefficients give i in mm/h with t in minutes. The defaults are illustrative only; use your local IDF curve.

Q(t) = C·i(t)·A·min(1, t/t_c). Shorter storms are more intense but only part of the catchment contributes; the peak is at t = t_c.

Comparison

Rational Method vs SCS Curve Number Method

Aspect	Rational method	SCS-CN method
Output	Peak discharge Q	Runoff depth and volume
Rainfall input	Intensity for duration t_c	Total storm depth
Catchment size	Small (≲ 80 ha / 200 ac)	Small to large (with routing)
Loss parameter	C (0–1)	CN (30–100)
Hydrograph	No (modified rational: approximate)	Yes, with a unit hydrograph
Typical use	Storm sewers, inlets, gutters, small culverts	Detention, floodplain studies, watershed models

Try the SCS Curve Number Runoff Calculator for the volume side of the same problem.

Avoid These

Common Mistakes in Rational Method Calculations

Mixing units

Using mm/h with acres, or forgetting /360 with hectares. Write the units next to every number.

Wrong duration

Reading the 60-minute intensity when t_c is 15 minutes can under-estimate Q by half or more.

One t_c formula for every flow path

Split the path into sheet, shallow concentrated and channel flow, and add the times.

Ignoring the size limit

Applying the method to a 5 km² rural catchment ignores rainfall variability and storage.

Area-weighting t_c or i

Only C is area-weighted. When combining sub-catchments downstream, use the longest t_c to the design point and ΣCA.

Partial-area effect

A small paved area near the outlet with a short t_c can produce a higher peak than the whole catchment. Check it.

Keep Learning

Related Tools and Guides

FAQ

Frequently Asked Questions

What is the rational method formula?

Q = C i A. In US units Q is in cfs, i in in/h and A in acres (the conversion factor 1.008 is taken as 1). In SI units Q (m³/s) = C i A / 360 with i in mm/h and A in hectares, or Q = 0.278 C i A with A in km². C is the dimensionless runoff coefficient.

What is the maximum drainage area for the rational method?

It is intended for small catchments. TxDOT limits it to about 200 acres (80 ha); many agencies use similar limits between roughly 20 and 200 acres. Larger areas violate the assumption of uniform rainfall over the whole catchment for a duration equal to the time of concentration, and storage effects become important.

Why is rainfall duration set equal to the time of concentration?

Peak flow occurs when the entire catchment contributes at once, which first happens after the time of concentration. Shorter storms are more intense but only part of the area contributes; longer storms cover the whole area but are less intense. For the rational method assumptions, duration = tc gives the maximum product i × contributing area.

What is a typical runoff coefficient for a residential area?

TxDOT gives C = 0.30–0.50 for single-family residential areas and 0.30–0.70 for apartment areas. Roofs are 0.75–0.95 and asphalt streets 0.85–0.95. Lawns range from about 0.05 on flat sandy soil to 0.35 on steep clay soil. Mixed catchments use an area-weighted composite C.

Can C be greater than 1?

No. C is the fraction of rainfall that becomes peak runoff. When a frequency factor is applied for 25- to 100-year storms (1.1 to 1.25), the product C × Cf should be capped at 1.0.

What is the difference between the rational method and the SCS curve number method?

The rational method gives a peak discharge for small catchments using rainfall intensity. The SCS curve number method gives runoff depth and volume from storm rainfall depth and needs a separate timing method for peaks. Use rational for storm drains and culverts on small sites; use SCS-CN or a unit hydrograph for volumes, detention and larger watersheds.

Sources

References

Texas Department of Transportation (TxDOT). Hydraulic Design Manual, Chapter 4, Section 11 (Time of Concentration) and Section 12 (Rational Method). txdot.gov

Chow, V. T., Maidment, D. R., & Mays, L. W. (1988). Applied Hydrology, Chapter 15. McGraw-Hill, New York.

ASCE & WPCF (1969). Design and Construction of Sanitary and Storm Sewers. ASCE Manual of Practice No. 37.

Kirpich, Z. P. (1940). Time of concentration of small agricultural watersheds. Civil Engineering, 10(6), 362.

Federal Aviation Administration (1970). Airport Drainage, Advisory Circular 150/5320-5B.

Kerby, W. S. (1959). Time of concentration for overland flow. Civil Engineering, 29(3), 174.

USDA Soil Conservation Service (1986). Urban Hydrology for Small Watersheds, TR-55.

NOAA National Water Center. NOAA Atlas 15 project status. water.noaa.gov/about/atlas15