Stefan-Boltzmann Radiation Calculator
The Stefan-Boltzmann law describes the total thermal radiation emitted by a surface. For a real surface with emissivity epsilon, radiating area A, and absolute temperature T, the radiated power is P = epsilon x sigma x A x T^4, where sigma = 5.670374419 x 10^(-8) W/(m^2 K^4) is the Stefan-Boltzmann constant (NIST CODATA 2018). This law is fundamental to thermal engineering, astrophysics, climate science, and building physics. Note that the T^4 dependence means that even modest temperature increases greatly increase radiation; conversely, cooling by radiation is more effective at high temperatures.
Stefan-Boltzmann law
P = epsilon × sigma × A × T⁴
sigma = 5.670374419 x 10^(-8) W/(m^2 K^4) (NIST CODATA 2018). epsilon is emissivity (dimensionless, 0 to 1). A is surface area in m^2. T is absolute temperature in Kelvin. P is total radiated power in Watts. Radiation flux = P/A = epsilon sigma T^4 (W/m^2).
Worked example: human body
- Average skin temperature: approximately 33 degrees C = 306 K.
- Body surface area: approximately 1.8 m^2.
- Emissivity of skin: approximately 0.98.
- P = 0.98 x 5.670 x 10^(-8) x 1.8 x 306^4 = approximately 492 W radiated.
- However, the body also absorbs radiation from surroundings. Net radiation loss at 20 degrees C room = much less than 492 W.
Frequently asked questions
What is the Stefan-Boltzmann law?
The Stefan-Boltzmann law states that the total energy radiated per unit area by a black body is proportional to the fourth power of its absolute temperature: P = sigma T^4 per unit area. For real surfaces with emissivity epsilon less than 1: P = epsilon sigma A T^4.
What is the Stefan-Boltzmann constant sigma?
The Stefan-Boltzmann constant sigma = 5.670374419 x 10^(-8) W/(m^2 K^4), as defined by NIST CODATA 2018. It can be derived from other fundamental constants: sigma = 2 pi^5 k^4 / (15 h^3 c^2).
What is emissivity epsilon?
Emissivity (epsilon, 0 to 1) is the ratio of a surface's thermal emission to that of a perfect black body at the same temperature. A perfect black body has epsilon = 1. Polished aluminum: about 0.05. Human skin: about 0.98. Oxidized iron: about 0.8.
Why does power scale with T^4?
The T^4 dependence arises from integrating the Planck distribution over all wavelengths. This means that doubling the absolute temperature increases radiated power by a factor of 2^4 = 16. Small temperature increases cause large changes in radiated power.
How is the Stefan-Boltzmann law used?
Applications include calculating heat loss from furnaces and ovens, estimating stellar luminosity from surface temperature (L = 4 pi R^2 sigma T^4), designing spacecraft thermal control systems, and analyzing climate feedback from Earth's radiation budget.
Official sources
- NIST CODATA 2018: Stefan-Boltzmann constant (NIST).
- OpenStax University Physics: 1.6 Mechanisms of Heat Transfer (Radiation).
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.