Stefan-Boltzmann law: every surface radiates thermal energy at a rate that climbs with the fourth power of its absolute temperature, P = σAεT⁴. This free calculator solves for the radiated power or the temperature, in any unit, and shows every step of the working.
The Stefan-Boltzmann law tells you how much thermal power a hot surface gives off as electromagnetic radiation. To calculate it, multiply four things together: the Stefan-Boltzmann constant σ (5.670374×10⁻⁸ W·m⁻²·K⁻⁴), the radiating area A, the emissivity ε, and the absolute temperature raised to the fourth power — P = σ·A·ε·T⁴. The answer is a power in watts.
There are three steps. First, decide whether you want the power or the temperature, and select it in the calculator’s Solve for menu. Second, enter the values you know: the surface area (square metres or square centimetres), the emissivity, and the temperature in kelvin. The emissivity is a fraction between 0 and 1 — use 1 for an ideal black body, or a measured value for a real surface. Third, read the answer with the worked steps, which show the formula, your numbers substituted in, and the result with units.
The single most common mistake is using the wrong temperature scale. The T⁴ term demands an absolute temperature, so values must be in kelvin (K = °C + 273.15). Because the dependence is to the fourth power, a small error in temperature becomes a large error in power. Keeping temperatures in kelvin — as this calculator requires — avoids the problem entirely.
To go the other way and find a surface’s temperature from its emitted power, the law rearranges to T = (P / σAε)^¼. This non-contact method underlies infrared thermometers and astronomical estimates of stellar surface temperatures. For the heat needed to warm a body in the first place, see the specific heat calculator; for the term itself, see the physics glossary.
A black-body radiator (ε = 1) has a surface area of 1 m² and is held at 1000 K. Its radiated power is P = σ·A·ε·T⁴ = 5.670374×10⁻⁸ × 1 × 1 × 1000⁴ ≈ 5.67×10⁴ W, about 56.7 kW. If the temperature is doubled to 2000 K, the T⁴ term grows by a factor of 2⁴ = 16, so the power leaps to roughly 907 kW — illustrating just how steeply thermal radiation rises with temperature.
The Stefan-Boltzmann law underpins the energy balance of stars and planets (including Earth’s climate), the design of furnaces, radiators and incandescent lamps, the calibration of infrared cameras and pyrometers, and the thermal management of spacecraft that can only shed heat by radiation.
The Stefan-Boltzmann law states that the thermal power radiated by a surface is proportional to the fourth power of its absolute temperature: P = σAεT⁴. Here σ is the Stefan-Boltzmann constant, A is the radiating area, ε is the emissivity and T is the temperature in kelvin. Because of the T⁴ term, doubling the absolute temperature increases the radiated power sixteenfold.
The Stefan-Boltzmann constant σ is 5.670374×10⁻⁸ W·m⁻²·K⁻⁴ (more precisely 5.670374419×10⁻⁸). It is a fixed quantity derived from the speed of light, the Planck constant and the Boltzmann constant, and it sets the absolute scale of thermal radiation.
Emissivity (ε) is a number between 0 and 1 that describes how efficiently a surface radiates compared with a perfect black body. A perfect black body has ε = 1, polished metals can be as low as 0.02, and most everyday painted or oxidised surfaces fall between 0.8 and 0.95. Multiply the black-body power by ε to get the real radiated power.
The T⁴ term uses absolute temperature, so the zero point must be absolute zero. Use kelvin, where K = °C + 273.15. Plugging in degrees Celsius gives a badly wrong answer because the law is not linear — a small absolute-temperature error is amplified to the fourth power.
Rearrange the law to T = (P / σAε)^¼. Divide the radiated power by σ, the area and the emissivity, then take the fourth root. Select “Temperature” in the calculator’s “Solve for” menu and it performs this rearrangement automatically — this is the basis of how a pyrometer or infrared thermometer reads temperature without contact.