Heat index is calculated using the NOAA Rothfusz regression equation, which takes air temperature in Fahrenheit and relative humidity as a percentage and returns a perceived temperature.
This article presents the full formula with labeled coefficients, a step-by-step manual calculation, a lookup chart organized by temperature band, and a clear explanation of when the formula is and is not valid. You can also check out our heat index calculator (it’s free) to calculate the optimal heat index for your environment.
Key Takeaways
- The NOAA Rothfusz regression equation uses nine polynomial terms combining air temperature and relative humidity to produce a heat index value in Fahrenheit.
- The Rothfusz formula is only valid when air temperature is at or above 80°F and relative humidity is at or above 40%, per the heat index equation reference. Below those thresholds, the National Weather Service uses a simpler Steadman-derived equation.
- A heat index at or above 103°F is classified by the National Weather Service as "Danger," with heat exhaustion likely and heat stroke possible on prolonged exposure.
- The formula assumes shaded conditions with low wind speed. Direct sunlight can add up to 15°F to the apparent temperature, making the calculated figure an underestimate in full-sun environments.
- OSHA's Heat Illness Prevention guidance identifies a heat index of 91°F or above as a trigger for employer action, including rest periods and water access requirements.
- High relative humidity is the primary driver of elevated heat index values because it slows sweat evaporation, reducing the body's ability to shed heat.
What Is Heat Index and Why Does Humidity Drive It
Heat index, also called apparent temperature or feels-like temperature, is the temperature the human body perceives when air temperature and relative humidity act together. The physiological mechanism is straightforward: the body cools itself through sweat evaporation, and high relative humidity slows that evaporation. When moisture cannot leave the skin efficiently, the body retains more heat than the air temperature alone would suggest.
This is why 90°F at 90% relative humidity feels significantly hotter than 90°F at 30% relative humidity, as explained by the heat index. The air temperature is identical. The perceived heat is not. The formula used to quantify this difference was developed as a regression fit to Robert Steadman's 1979 physiological research on human thermal comfort, which related body heat dissipation rates to ambient temperature and humidity conditions.
Understanding dew point vs humidity also helps here. Dew point is a closely related measure of atmospheric moisture, but the heat index formula uses relative humidity, expressed as a humidity percentage, as its direct input.
Heat Index vs. Actual Temperature
Heat index and actual temperature measure two different things:
- Actual temperature: the measured air temperature, read directly from a thermometer.
- Heat index: not a measurement at all, it's a calculated index of perceived heat based on how the body experiences the combination of temperature and humidity.
- The gap between them can be large: at 90°F and 90% relative humidity, for example, the heat index can exceed 120°F.
The Rothfusz formula is what quantifies that gap between actual and apparent temperature.
The NOAA Heat Index Formula (Rothfusz Regression Equation)
The heat index formula published by the National Weather Service heat index equation is the standard used for public forecasts and heat advisories across the United States. It is a polynomial regression, not a derivation from physical principles. The equation was fitted to Steadman's original heat index tables to produce a continuous function that returns apparent temperature in Fahrenheit for any combination of temperature and humidity within the valid input range.
The Rothfusz regression equation is written as:
HI = c1 + c2T + c3R + c4TR + c5T² + c6R² + c7T²R + c8TR² + c9T²R²
The variables are:
- T = dry-bulb air temperature in degrees Fahrenheit
- R = relative humidity as a percentage (0 to 100)
- HI = heat index (apparent temperature) in degrees Fahrenheit
This heat index formula is what any reliable heat index calculator is applying under the surface. NOAA and the National Weather Service use it as the basis for all official heat index products.
What Each Coefficient Represents
The nine coefficients in the Rothfusz regression equation are empirically fitted constants, not physical constants. Their values were determined by fitting the polynomial to Steadman's tabulated data. They carry no independent physical meaning beyond that regression fit.
The NWS-published coefficient values are:
- c1 = −42.379
- c2 = 2.04901523
- c3 = 10.14333127
- c4 = −0.22475541
- c5 = −0.00683783
- c6 = −0.05481717
- c7 = 0.00122874
- c8 = 0.00085282
- c9 = −0.00000199
The equation produces output in Fahrenheit. Celsius inputs and outputs require unit conversion, covered in the next section.
How to Calculate Heat Index in Celsius
The Rothfusz formula operates in Fahrenheit only. To calculate heat index from a Celsius temperature, convert the input first using F = C × 9/5 + 32. Apply the full formula to the converted value. Then convert the result back to Celsius using C = (F − 32) × 5/9. Relative humidity is dimensionless and requires no conversion at any step.
Step-by-Step Worked Example: Manual Heat Index Calculation
No competitor page walks through this calculation with actual numbers. That is the gap this section fills. Using T = 96°F and R = 65%, here is the full manual calculation:
- c1 = −42.379
- c2 × T = 2.04901523 × 96 = 196.705
- c3 × R = 10.14333127 × 65 = 659.317
- c4 × T × R = −0.22475541 × 96 × 65 = −1,402.474
- c5 × T² = −0.00683783 × 9,216 = −63.013
- c6 × R² = −0.05481717 × 4,225 = −231.503
- c7 × T² × R = 0.00122874 × 9,216 × 65 = 735.966
- c8 × T × R² = 0.00085282 × 96 × 4,225 = 345.883
- c9 × T² × R² = −0.00000199 × 9,216 × 4,225 = −77.485
Sum all terms:
−42.379 + 196.705 + 659.317 − 1,402.474 − 63.013 − 231.503 + 735.966 + 345.883 − 77.485 = 121.0°F
At 96°F and 65% relative humidity, the NOAA heat index formula gives a heat index of approximately 121°F. This falls squarely in the "Danger" category on the National Weather Service scale. You can replicate this calculation in a spreadsheet using a single formula cell with T and R as adjacent cell references.
Using a Heat Index Calculator vs. Manual Calculation
A heat index calculator applies the same Rothfusz formula automatically and is appropriate for quick lookups. Manual calculation matters when you are implementing the formula in a data pipeline, spreadsheet model, or field instrument where the output cannot be independently verified. Any reliable heat index calculator should return approximately 121°F for 96°F and 65% relative humidity. A discrepancy indicates either a different formula variant or an input unit mismatch.
Heat Index Reference Chart (Temperature × Relative Humidity)
The values below are drawn from the National Weather Service heat index chart and reflect shaded conditions with low wind speed. Direct sunlight can add up to 15°F to the apparent temperature, so treat these figures as conservative estimates for full-sun environments. The chart covers the range where the Rothfusz formula is valid: 80°F and above, 40% relative humidity and above. A full NWS heat index chart with the complete grid is available at the National Weather Service website.
80°F to 89°F Band
- 80°F at 40% RH: approximately 80°F heat index
- 80°F at 50% RH: approximately 81°F, according to the heat index chart
- 80°F at 60% RH: approximately 82°F, according to the heat index
- 80°F at 70% RH: approximately 83°F on the heat index
- 80°F at 80% RH: approximately 84°F on the heat index
- 80°F at 90% RH: approximately 86°F, according to the heat index
- 85°F at 40% RH: approximately 85°F on the heat index
- 85°F at 50% RH: approximately 88°F, per the heat index scale
- 85°F at 60% RH: approximately 90°F heat index
- 85°F at 70% RH: approximately 93°F on the heat index scale
- 85°F at 80% RH: approximately 97°F heat index
- 85°F at 90% RH: approximately 102°F heat index
90°F to 99°F Band
- 90°F at 40% RH: approximately 91°F heat index
- 90°F at 50% RH: approximately 95°F heat index
- 90°F at 60% RH: approximately 100°F heat index
- 90°F at 70% RH: approximately 106°F heat index
- 90°F at 80% RH: approximately 113°F heat index
- 90°F at 90% RH: approximately 122°F heat index
- 95°F at 40% RH: approximately 97°F heat index
- 95°F at 50% RH: approximately 101°F heat index
- 95°F at 60% RH: approximately 114°F heat index
- 95°F at 70% RH: approximately 124°F heat index
- 95°F at 80% RH: approximately 133°F heat index
100°F and Above
- 100°F at 40% RH: approximately 109°F heat index
- 100°F at 50% RH: approximately 118°F heat index
- 100°F at 60% RH: approximately 129°F heat index
- 100°F at 70% RH: approximately 143°F heat index
- 105°F at 40% RH: approximately 121°F heat index
- 105°F at 50% RH: approximately 133°F heat index
- 105°F at 60% RH: exceeds 149°F heat index
At high temperatures combined with high relative humidity, apparent temperature values can exceed 130°F, placing conditions firmly in the Extreme Danger category.
When the Rothfusz Formula Is Invalid
The Rothfusz regression equation was fitted to data in the high-temperature, high-humidity range. Outside that range, the polynomial produces outputs that diverge significantly from Steadman's original physiological tables. The National Weather Service restricts the formula to inputs where T is at or above 80°F and R is at or above 40%. Applying it below these thresholds does not return an error, but it does return an unreliable result.
The NWS specifies two additional adjustment conditions for the Rothfusz output. If relative humidity is below 13% and air temperature falls between 80°F and 112°F, a correction factor is subtracted from the result. If relative humidity exceeds 85% and air temperature falls between 80°F and 87°F, a correction factor is added. These adjustments account for ranges where the base polynomial overshoots or undershoots the tabulated values. Full sun and still air both push the real feel higher than the number the equation returns; direct sunlight alone adds up to approximately 15°F beyond what the polynomial captures.
The NWS validity rules as bullets:
- If T is below 80°F, use the simpler Steadman-derived equation instead of Rothfusz.
- If the simple equation's result averaged with T is below 80°F, report that average as the heat index.
- If RH is below 13% and T is between 80°F and 112°F, apply the low-humidity subtraction adjustment.
- If RH is above 85% and T is between 80°F and 87°F, apply the high-humidity addition adjustment.
Low-Temperature Adjustment: Below 80°F
When air temperature is below 80°F, the NWS uses this simpler Steadman-derived equation:
HI = 0.5 × (T + 61.0 + [(T − 68.0) × 1.2] + (RH × 0.094))
This equation is linear rather than polynomial. It is less precise than the Rothfusz formula but avoids the instability the polynomial produces at low-temperature inputs. If the result of this equation averaged with the actual air temperature falls below 80°F, that average is reported as the heat index.
Humidity Adjustment Conditions
Two correction factors apply to Rothfusz outputs at the edges of its valid humidity range. For low humidity (RH below 13%, T between 80°F and 112°F), the NWS subtracts the quantity [(13 − RH) / 4] × SQRT[(17 − |T − 95|) / 17] from the Rothfusz result. For high humidity (RH above 85%, T between 80°F and 87°F), the NWS adds [(RH − 85) / 10] × [(87 − T) / 5] to the result. Both corrections are documented in the NWS Heat Index Equation technical note.
Heat Index Danger Levels and Occupational Thresholds
Knowing the calculated heat index value matters only if you know what that number means in practice. The National Weather Service classifies heat index into four tiers, and OSHA maps those tiers to specific employer obligations. Understanding the connection between the formula output and these regulatory thresholds is what turns a calculated number into an actionable safety input.
The four NWS heat index categories are:
- Caution (80°F to 90°F): Fatigue is possible with prolonged exposure and physical activity. Heat-related illness is unlikely but possible in sensitive individuals.
- Extreme Caution (91°F to 103°F): Heat cramps and heat exhaustion are possible. Outdoor safety planning should begin here.
- Danger (103°F to 124°F): Heat exhaustion is likely. Heat stroke is possible with prolonged exposure or strenuous activity.
- Extreme Danger (above 125°F): Heat stroke is imminent. This level presents life-threatening risk with rapid onset.
Heat stroke is a medical emergency. It can develop rapidly at high heat index values and can follow heat exhaustion if intervention does not occur. The NWS classifications are drawn from NOAA heat index public guidance.
OSHA Heat Illness Prevention Requirements
OSHA's Heat Illness Prevention program uses heat index as a component of its risk assessment framework for outdoor and indoor workers. At heat index levels of 91°F or above, employers are generally expected to implement preventive measures. These include rest periods, water access, and shade provisions. At 103°F and above, OSHA considers conditions "high to very high risk," requiring more structured intervention and acclimatization protocols. The OSHA Heat Safety Tool provides employer-facing guidance for assessing risk by location, date, and activity level. OSHA's framework does not set a single binding numerical threshold, but heat index is a primary input to the risk tier determination.
Heat Index in Indoor Industrial Environments
Heat index applies indoors as well as outdoors. Manufacturing plants, warehouses, foundries, and facilities with significant heat-generating equipment can reach temperature and relative humidity combinations that produce elevated apparent temperatures even without solar exposure.
Understanding how humidity is measured in indoor environments is a prerequisite for any credible heat index monitoring program. Humidity control is one facility-level intervention that can influence indoor heat index by moderating the relative humidity input to the heat index equation.
Why Precise Humidity Control Matters for Indoor Heat Index Management
Indoor heat index is a function of two variables: air temperature and relative humidity. Facility operators can influence air temperature through HVAC design, but relative humidity requires dedicated humidity control systems. When the RH input to the heat index equation is unstable or inaccurate, any heat index figure derived from it is correspondingly unreliable, which undermines the value of an OSHA-compliant heat illness prevention program that depends on it.
Precision in humidity management matters here specifically because the heat index formula is non-linear. At elevated temperatures, small changes in relative humidity produce disproportionately large changes in apparent temperature. A facility running at 90°F with RH that oscillates between 55% and 75% is not experiencing a stable 65% heat index environment. It is experiencing a heat index that swings by more than 15°F depending on the time of measurement.
How Humidity Precision Affects Indoor Apparent Temperature
Maintaining stable relative humidity is a prerequisite for accurate indoor heat index assessment. Precision humidity control systems that hold RH within plus or minus 1 to 2% keep the humidity input to the heat index calculation stable enough to be meaningful. Smart Fog's industrial humidification systems are engineered to maintain relative humidity at that precision level continuously, which means the RH figure fed into any heat index calculation reflects actual conditions rather than a wide average. This level of control is directly relevant for facilities running OSHA-aligned heat illness prevention programs that treat heat index as a primary risk input.
Industrial Humidification for Facilities with Heat Stress Monitoring
Facilities with indoor heat stress concerns need humidity control that adds moisture without compounding heat stress through surface wetting or excess condensation. Smart Fog systems produce self-evaporating droplets from an equal-sized droplet grid. This technology adds moisture to the air without wetting surfaces, equipment, or workers under proper system design.
The result is stable, measurable relative humidity in occupied industrial environments. Note that "non-wetting" applies under proper system design. Direct exposure to the fog stream will wet surfaces.
Maintenance intervals for Smart Fog systems extend up to every two years, with no moving parts in the humidification process, supporting continuous operation in demanding environments. For facilities evaluating the role of evaporative cooling alongside humidification, the distinction between cooling effect and humidity control is worth examining separately.
Final Thoughts
Heat index is a calculated value, not a measured one. Getting it right requires accurate inputs and a clear understanding of where the formula is and is not valid. Reach for the Rothfusz regression equation once air temperature clears 80°F and relative humidity clears 40%; below either threshold, a different equation takes over. In direct sunlight, the calculated figure understates the actual thermal load by up to 15°F.
For indoor environments where both air temperature and relative humidity are variables that can be actively managed, the accuracy of your heat index assessment depends directly on the accuracy of your humidity measurement and control. Unstable or imprecise RH makes indoor heat index monitoring an estimate at best and a liability gap at worst.
If your facility operates a heat illness prevention program that relies on heat index as a risk input, contact Smart Fog engineers to discuss humidity control system options for your specific environment and operational requirements.
FAQ
What is the formula for calculating the heat index?
The heat index formula is the NOAA Rothfusz regression equation: HI = c1 + c2T + c3R + c4TR + c5T² + c6R² + c7T²R + c8TR² + c9T²R², where T is air temperature in Fahrenheit, R is relative humidity as a percentage, and c1 through c9 are nine empirically fitted coefficients published by the National Weather Service. The equation returns apparent temperature in Fahrenheit and is the standard formula used for all official NOAA heat index forecasts and advisories.
How do you calculate heat index from temperature and humidity?
To calculate heat index, you substitute your air temperature in Fahrenheit and your relative humidity percentage into the Rothfusz regression equation and sum all nine polynomial terms. For example, at 96°F and 65% relative humidity, the formula returns approximately 121°F. Below 80°F air temperature or 40% relative humidity, this polynomial isn't the right tool at all; the National Weather Service switches to a simpler linear equation in that range to avoid unreliable results.
What heat index level is considered dangerous?
The National Weather Service classifies a heat index of 103°F to 124°F as "Danger," meaning heat exhaustion is likely and heat stroke is possible with prolonged exposure. Above 125°F, the classification is "Extreme Danger," with heat stroke considered imminent. OSHA identifies heat index at or above 103°F as a "high to very high risk" condition, requiring structured employer intervention including rest periods and acclimatization protocols.
What is the difference between heat index and actual temperature?
Heat index is a calculated index of how hot the air feels to the human body, based on the combined effect of air temperature and relative humidity. Actual temperature is the measured dry-bulb air temperature with no adjustment for humidity. The gap between the two can be large: at 90°F and 90% relative humidity, the heat index can exceed 120°F, more than 30°F above the measured air temperature.
How does humidity affect the heat index?
High relative humidity slows the evaporation of sweat from the skin, which reduces the body's ability to shed heat. When sweat cannot evaporate efficiently, the body retains more thermal energy than the air temperature alone would cause. This is why heat index rises steeply as humidity percentage increases, especially above 80°F. At constant temperature, a shift from 40% to 80% relative humidity can add 20°F or more to the apparent temperature.
When does the NOAA heat index formula give inaccurate results?
The Rothfusz regression equation gives unreliable results when air temperature is below 80°F or relative humidity is below 40%. The equation was fitted to high-temperature, high-humidity data, so it diverges from Steadman's physiological tables at low-temperature or low-humidity inputs. The National Weather Service applies a simpler linear formula below the 80°F threshold and adds humidity-based correction factors when relative humidity is below 13% or above 85% within specific temperature ranges.
How do you calculate heat index manually without a calculator?
To calculate heat index manually, substitute your temperature in Fahrenheit and your relative humidity percentage into each of the nine terms of the Rothfusz regression equation, compute each term individually, and sum the results. For a worked example: at T = 96°F and R = 65%, the nine terms sum to approximately 121°F. Each term can be computed with a basic calculator or in a spreadsheet cell. The full NWS coefficients are c1 = −42.379, c2 = 2.04901523, c3 = 10.14333127, c4 = −0.22475541, c5 = −0.00683783, c6 = −0.05481717, c7 = 0.00122874, c8 = 0.00085282, and c9 = −0.00000199.
What does OSHA require when the heat index reaches 91°F or higher?
OSHA's Heat Illness Prevention guidance treats a heat index of 91°F or above as a "moderate risk" condition requiring employer action. At this level, employers are generally expected to provide water, rest breaks, and shade or cooling access for workers. At 103°F and above, OSHA classifies conditions as "high to very high risk," requiring more structured measures including acclimatization schedules for new or returning workers. There's no single hard number in OSHA's rulebook here; heat index simply feeds directly into which risk tier, and which obligations, apply.
