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Stefan-Boltzmann law

A black body with 1 m² of surface at 300 K radiates 459.300327939 W. Enter the area and the temperature in kelvin to get the total radiated power, with emissivity fixed at 1 for a perfect emitter. Free in your browser.

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Worked example

If you enter

Area: 1
Creativity: 300

You get

Sigma: 0.0000000567 W
Power: 459.300327939
Precision: 10

The Stefan-Boltzmann law is the closed-form statement that an ideal black body radiates a total thermal power proportional to the fourth power of its absolute temperature. This page evaluates that identity: P equals sigma times A times T to the fourth, with emissivity fixed at one because a black body is a perfect emitter. Supply the radiating surface area and the temperature on the kelvin scale; when area is in square metres the result is the radiated power in watts, using the fixed CODATA Stefan-Boltzmann constant sigma equals 5.670374419 times ten to the minus eight watts per square metre per kelvin to the fourth. Zero area or zero kelvin is valid and yields zero watts. The engine is pure deterministic arithmetic: no network, no sampling, and no clocks.

What the Stefan-Boltzmann law says for an ideal black body

Josef Stefan measured that the total energy leaving a hot body grew with the fourth power of temperature; Ludwig Boltzmann later derived the same T-to-the-fourth scaling from thermodynamics and the radiation field. For an ideal black body the modern statement is compact: the total radiated power P equals sigma times the surface area A times the fourth power of the absolute temperature T. Sigma is a universal constant, not a material property.

A black body absorbs every incident photon and re-emits as a perfect thermal radiator, so its emissivity is identically one and does not appear as a free parameter on this page. The T-to-the-fourth dependence is the whole pedagogical point: doubling kelvin temperature multiplies radiated power by sixteen, which is why a small error in T dominates a radiation homework check and why stellar photospheres outshine household heaters by many orders of magnitude even at comparable area.

Absolute temperature must be in kelvin because the law is written on the thermodynamic scale that starts at zero. A value below zero kelvin is physically meaningless in this model and is rejected rather than silently folded into a fourth-power of a negative number. Zero kelvin is allowed and yields zero power, matching the statement that a body at absolute zero emits no thermal radiation.

This calculator returns the normalized area and temperature, the fixed sigma it used, the computed power in watts for SI inputs, the rounding precision, a method tag, a short formula reminder, and the unit W. It does not look up grey-body emissivities, add a surrounding T_env to the fourth for net exchange, or expand a Planck spectrum: those are neighbouring models. The question here is only the total power leaving a black surface of known area at a known kelvin temperature.

Where a Stefan-Boltzmann law calculator earns its keep

Students meet P equals sigma A T to the fourth when a thermodynamics chapter introduces thermal radiation, when a black-body problem gives area and kelvin temperature and asks for watts, or when a contest item wants the T-to-the-fourth scaling without a full radiative-transfer code. Teachers can pre-generate answer keys for modest area and temperature without trusting opaque apps that silently treat Celsius as kelvin or slip an emissivity of 0.9 into a problem that specified a black body.

Astronomy and stellar-structure primers use the same identity as a first estimate of the luminosity of a spherical photosphere once radius and effective temperature are known: four pi R squared plays the role of A, and the result is still watts. Spacecraft thermal sketches and furnace first-order checks use the black-body law when only area and surface temperature are available before view factors, coatings, and convection are modelled.

Developers call the contract from scripts to seed deterministic test vectors for radiation modules, to normalize user-entered black-body loads, or to satisfy programming-challenge statements that demand watts rather than a hand-waved estimate. Keep requests inside the published ceilings, pin the rounded power in tests, and you get reproducible black-body watts for homework, LMS plugins, and edge workers without standing up a heat-transfer stack of your own.

What you can do with it

Homework and black-body labs

Compute P = σAT⁴ from area and kelvin temperature when a worksheet specifies an ideal black body rather than a grey surface.

First-order thermal sketches

Turn a known radiating area and surface temperature into watts before adding coatings, view factors, or a surrounding environment.

FAQ

How does this Stefan-Boltzmann law calculator define radiated power?

For an ideal black body, P = σ · A · T⁴, with σ = 5.670374419e-8 W/(m²·K⁴). Area A is in square metres and T is absolute temperature in kelvin; the result is in watts. Emissivity is fixed at 1.

What happens if temperature is negative kelvin?

Absolute temperature cannot be below 0 K; the Stefan-Boltzmann law is undefined for negative kelvin in this calculator, and the failure is not billed.

Do I need to supply emissivity?

No. This landing is the black-body law, so emissivity is identically one and is not an input. Grey-body problems that need a fractional ε are a different search intent.

Does the tool convert Celsius or Fahrenheit to kelvin?

Convert from other scales before calling the calculator.

What happens to the radiated power if the temperature doubles?

It becomes sixteen times larger, because power grows with the fourth power of the kelvin temperature. That is why a small error in the temperature dominates a radiation exercise far more than an error in the area.

Related tools

For developers — API access

Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.

API endpoint

POSThttps://api.kit.forhosting.com/phys/stefan-boltzmann-law

Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.

Call it from your stack

curl -X POST https://api.kit.forhosting.com/phys/stefan-boltzmann-law \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"area":1,"temperature":300}'

const res = await fetch("https://api.kit.forhosting.com/phys/stefan-boltzmann-law", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "area": 1,
    "temperature": 300
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/phys/stefan-boltzmann-law",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "area": 1,
    "temperature": 300
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/stefan-boltzmann-law", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"area":1,"temperature":300}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"area":1,"temperature":300}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/stefan-boltzmann-law", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "area": 1,
  "temperature": 300
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "phys.stefan_boltzmann_law",
  "status": "queued",
  "_links": {
    "result": "/tasks/tsk_…/result"
  }
}

The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.

Pricing

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Limits

`max_abs_area`	1000000000000000
`max_abs_temperature`	1000000000000

Errors

HTTP	Code	Meaning
`401`	`unauthorized`	Missing or invalid API key.
`402`	`insufficient_balance`	Your balance doesn't cover the task price.
`404`	`unknown_type`	That task type doesn't exist.
`429`	`rate_limited`	Too many requests. Use the webhook instead of polling.

Technical notes

Negative kelvin is rejected with invalid_input, because thermodynamic temperature cannot fall below absolute zero.

Input contract, kelvin temperature, and how P is kept reproducible

Zero area is allowed and returns zero power. Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; empty strings, Infinity, NaN, and non-numeric tokens are rejected. Magnitude ceilings on area and temperature stop overflow into non-finite power. Failed validation is not billed. POST a JSON body with two required finite numbers: area, the radiating surface in square metres, and temperature, the absolute temperature in kelvin. Aliases A, surface_area, and surface are accepted for area; aliases T, temp, kelvin, and K are accepted for temperature so spreadsheets and typed forms reuse the same contract. Area must be non-negative: a negative value fails with invalid_input naming the field, because a negative surface is not a radiating area in this model. Temperature must be non-negative: negative kelvin fails with invalid_input and a message that names the field and states that the Stefan-Boltzmann law is undefined for temperatures below absolute zero. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose, defaulting to ten and capped at fifteen so JSON goldens stay stable across runtimes. Pair the free browser run with the priced API when a pipeline must assert on a known black-body power without re-deriving sigma by hand.

Integration use cases

Deterministic radiation fixtures

Generate reproducible black-body powers for unit tests, teaching demos, and CI assertions on the Stefan-Boltzmann T⁴ law.

Technical questions

What does the Stefan-Boltzmann law calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

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