Blackbody radiation calculator
Radiated power from temperature and area, and the wavelength where it peaks.
What is the Stefan–Boltzmann law?
A surface at absolute temperature T radiates the power P = εσAT⁴, where A is its area, ε its emissivity (1 for an ideal blackbody, between 0 and 1 for real surfaces) and σ = 5.670374 × 10⁻⁸ W/(m²·K⁴). The calculator computes σ from the exact SI constants h, c and k as 2π⁵k⁴ ÷ (15h³c²), which matches NIST CODATA 2022; OpenStax rounds it to 5.67 × 10⁻⁸. Because of the fourth power, a star at 25,000 K radiates about 4,800 times more per square meter than one at 3,000 K (OpenStax University Physics Vol. 3, Example 6.2). Temperatures must be in kelvins; the calculator converts °C and °F for you.
How much heat does a person radiate?
A body also absorbs radiation from its surroundings, so the net loss is εσA(T⁴ − Tₛ⁴). OpenStax University Physics Vol. 2, Section 1.6: skin at 33.0 °C, a room at 22.0 °C, 1.50 m² of skin and ε = 0.97 (skin in the infrared) give a net loss of about 99 W. That is the pre-filled example; with the exact 273.15 K offsets it comes to 98.7 W, since the book rounds to 306 K and 295 K. Enter the surroundings to get the net power; leave it blank for the emitted power alone.
What is Wien’s displacement law?
The wavelength where a blackbody spectrum peaks is λmax = b ÷ T, with b = 2.897771955 × 10⁻³ m·K (OpenStax rounds it to 2.898 × 10⁻³). It comes from hc ÷ (k x), where x = 4.96511… solves (x − 5)eˣ + 5 = 0, so the calculator derives b from the exact constants too. The Sun’s effective temperature, 5772 K (IAU 2015 nominal value), peaks at 502 nm, in visible light; a body at 37 °C peaks near 9.3 µm, in the infrared. Region names follow OpenStax University Physics Vol. 2, Section 16.5: visible light is about 400–750 nm and ultraviolet runs from 400 nm down to about 10 nm; those limits are approximate.
Why isn’t the peak frequency just c divided by the peak wavelength?
A blackbody spectrum can be written per unit wavelength or per unit frequency, and the two curves peak at different places. Per unit frequency the peak is at f = 5.8789 × 10¹⁰ Hz/K × T (the Wien frequency displacement constant, NIST CODATA 2022), so for the Sun it is about 339 THz, which corresponds to roughly 880 nm, not 502 nm. The calculator shows the peak wavelength from Wien’s law and notes the frequency peak separately.
Example: how much power does the Sun radiate?
Treat the Sun as a blackbody with ε = 1, the IAU 2015 nominal effective temperature T = 5772 K and nominal radius R = 6.957 × 10⁸ m. Its area is 4πR² = 6.082 × 10¹⁸ m², so P = σAT⁴ = 3.828 × 10²⁶ W, the IAU nominal solar luminosity. Enter those three numbers in the calculator to reproduce it. These IAU values are conversion constants chosen to agree with each other, not measurements made at the last digit.
Worked example: a glowing heating element
A flat surface of 0.02 m² at 800 °C has ε = 0.9. Convert to kelvins first: 800 + 273.15 = 1073.15 K. Then P = εσAT⁴ = 0.9 × 5.670374×10⁻⁸ × 0.02 × 1073.15⁴ = 1354 W, emitted from one face. The peak wavelength is b ÷ T = 2.70 µm, in the infrared, so most of the output is invisible heat rather than light. Doubling the absolute temperature multiplies the power by 2⁴ = 16, but doubling a Celsius reading does not, because the law needs kelvins.
Worked example: temperature from a measured peak
Wien’s law can be turned around: T = b ÷ λmax. A source whose spectrum peaks at 580 nm has T = 2.897771955×10⁻³ ÷ 580×10⁻⁹ = 4996 K. For comparison, objects at room temperature (293.15 K) peak at 9.9 µm, far from visible light, so a room-temperature object does not glow. The estimate works for objects that behave like blackbodies, not for gases with emission lines.
Common mistakes with blackbody calculations
Entering a temperature in °C or °F where kelvins are required, which makes the power far too small (the calculator converts if you choose the unit). Leaving out the emissivity for a real surface, which overstates the power for shiny metals. Forgetting that the area is the radiating area, which for a flat plate that emits from both faces is twice the plate area. Reading the net power as the emitted power: a body in a room also absorbs radiation, so its net loss is smaller.
Sources and related calculators
Sources: OpenStax University Physics Volume 2, Section 1.6 (heat transfer, radiation) and Section 16.5 (the electromagnetic spectrum); OpenStax University Physics Volume 3, Chapter 6 (blackbody radiation); NIST CODATA 2022 values of the Stefan–Boltzmann and Wien constants; IAU 2015 Resolution B3 for the nominal solar values. Related: heat loss calculator, heat conduction calculator, star magnitude calculator and specific heat calculator.