Stefan-Boltzmann calculator
A perfect black body of 1 m² at 300 K radiates 459.300327939 W. Enter the surface area, the emissivity and the absolute temperature to get the radiated power, using the fixed Stefan-Boltzmann constant. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Area
- 1
- Emissivity
- 1
- Creativity
- 300
- Sigma
- 0.0000000567 W
- Power
- 459.300327939
- Precision
- 10
This Stefan-Boltzmann calculator finds the thermal power radiated by a surface when you know its area, emissivity, and absolute temperature in kelvin. The defining relation is the closed form P equals epsilon times sigma times A times T to the fourth, which is the standard introductory model used in thermodynamics homework, black-body problems, and first-order heat-loss checks. With area in square metres and temperature in kelvin, the result is reported 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. Emissivity of one models an ideal black body; real materials use a fractional value you supply. Zero area or zero kelvin is valid and yields zero power; negative kelvin is rejected because absolute temperature cannot fall below zero in this model. The engine is fully deterministic arithmetic: no network calls, no sampling, and no date-dependent shortcuts.
Where radiated-power checks fit in study, design, and software
Students meet P equals epsilon sigma A T to the fourth when a textbook introduces thermal radiation, when a black-body problem asks how much power leaves a known surface at a stated kelvin temperature, or when a contest problem gives area, emissivity, and T and asks for watts without a full radiative-transfer simulation. Teachers can pre-generate answer keys for modest area and temperature values without trusting opaque calculator apps that silently switch between celsius and kelvin or confuse emissivity with absorptivity.
Developers call the same contract from scripts to seed deterministic test vectors for thermal simulators, to normalize user-entered radiation loads before a process sketch, or to satisfy programming-challenge statements that demand watts rather than a hand-waved estimate. Process and product first-order checks use the Stefan-Boltzmann law when only area, a known emissivity, and surface temperature are available before view factors, multi-surface exchange, or convection are modeled.
What you can do with it
Homework and lab checks
Compute P = εσAT⁴ from area, emissivity, and absolute temperature before grading radiation worksheets or black-body labs.
Process and tutoring software
Normalize user-entered area, ε, and kelvin temperature into watts for heat-loss sketches, tutoring UIs, and interactive lessons.
FAQ
What does the Stefan-Boltzmann calculator cost?
It is free in your browser on this page.
How is radiated power defined?
By the Stefan–Boltzmann law, P = ε · σ · A · T⁴, where A is surface area, ε is emissivity, T is absolute temperature in kelvin, and σ = 5.670374419e-8 W/(m²·K⁴). With SI units, P is in watts.
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.
Does this tool convert Celsius or Fahrenheit to kelvin?
Convert from other scales before calling the calculator.
Why are results rounded to a precision?
Default precision is 10 decimal places; you can set precision from 0 to 15.
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
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/thermo/stefan-boltzmann \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"area":1,"emissivity":1,"temperature":300}'const res = await fetch("https://api.kit.forhosting.com/thermo/stefan-boltzmann", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"area": 1,
"emissivity": 1,
"temperature": 300
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/thermo/stefan-boltzmann",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"area": 1,
"emissivity": 1,
"temperature": 300
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/thermo/stefan-boltzmann", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"area":1,"emissivity":1,"temperature":300}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"area":1,"emissivity":1,"temperature":300}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/thermo/stefan-boltzmann", 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,
"emissivity": 1,
"temperature": 300
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "thermo.stefan_boltzmann",
"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
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_abs_area | 1000000000000000 |
max_abs_emissivity | 1000000 |
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
What the Stefan-Boltzmann law means and what the response contains
A surface at absolute temperature T emits thermal radiation whose power scales with the fourth power of temperature. The Stefan-Boltzmann law states that the total radiated power is P equals epsilon times sigma times A times T to the fourth. Area A is the radiating surface area, epsilon is the emissivity (one for a perfect black body, typically between zero and one for real materials), and sigma is a universal constant. Zero area or zero kelvin is valid and yields zero power when the other factors are finite. Absolute temperature must be in kelvin: the law is written on the thermodynamic temperature scale, so a negative value is physically meaningless and is rejected. This calculator returns the normalized inputs area, emissivity, and temperature, the fixed sigma it used, the computed power, the precision used, a method tag stefan_boltzmann, a short formula reminder, and a unit hint W for SI inputs. There is no automatic material lookup, no surrounding-environment term such as T_env to the fourth for net exchange, and no Monte Carlo sampling: the engine either evaluates the product inside the supported numeric range or rejects the input with invalid_input. Rejected cases include a missing area, emissivity, or temperature, non-finite values, magnitudes above the documented ceilings, and any temperature that is negative kelvin, because absolute temperature cannot be below zero kelvin in this calculator.
How area, emissivity, and temperature are read and how P stays reproducible
Zero area is allowed and returns zero power. You choose the magnitude of epsilon; one means black-body emission, while painted metals, ceramics, and biological surfaces often use tabulated values between zero and one, but the tool never invents a material default and only rejects negative emissivity. Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces and non-numeric tokens are rejected. The preferred field for surface area is area, but aliases A, surface_area, and surface are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. Area must be finite and non-negative: a negative value fails with invalid_input and a message that names the field and explains that radiated power is undefined for negative surface area in this calculator, so a bad cell does not become a silent nonsense result. The preferred field for emissivity is emissivity, with aliases e, eps, and epsilon so textbook and engineering forms map cleanly. The preferred field for absolute temperature is temperature, with aliases T, temp, kelvin, and K. Temperature must be finite and non-negative: negative kelvin fails with invalid_input because the Stefan-Boltzmann law is undefined for temperatures below absolute zero in this calculator. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose for the numeric outputs, defaulting to ten and capped at fifteen so JSON goldens stay stable. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the fourth-power formula yourself.
Where radiated-power checks fit in study, design, and software
Product and education tools use the endpoint when a learner types area, epsilon, and kelvin temperature and the UI must show the radiated power. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with other thermo or mechanics helpers when you need related energy steps, and with math endpoints when you are building a small lab of free, deterministic physics calculators.
Integration use cases
Deterministic test fixtures
Generate reproducible radiated powers for unit tests, thermal demos, and CI assertions on the Stefan–Boltzmann law.