Temperature from radiated power
A surface of 1 m² with emissivity 1 that radiates 459.300327939 W is at 300 K. Enter the radiated power, the area and the emissivity to recover the temperature in kelvin for a black body or a real material. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Power
- 459.300327939
- Area
- 1
- Emissivity
- 1
- Sigma
- 0.0000000567 K
- Creativity
- 300
- Precision
- 10
This temperature from radiated power calculator finds the absolute temperature of a thermal radiator when you already know how many watts leave the surface, how large that surface is, and what emissivity to assign. The defining relation is the inverted Stefan–Boltzmann law: absolute temperature T equals the fourth root of power P divided by the product of emissivity, the Stefan–Boltzmann constant sigma, and area A. With power in watts and area in square metres the result is kelvin, using the fixed CODATA 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 paints, metals, ceramics, and biological surfaces use a positive fraction you supply. Zero power is valid and yields zero kelvin when area and emissivity are positive; negative power, zero area, or zero emissivity are rejected because the inverted law is undefined for those cases. The engine is pure deterministic arithmetic with no network calls, no sampling, and no date-dependent shortcuts.
How temperature follows from radiated power
Thermal radiation from a surface scales with the fourth power of absolute temperature. The forward Stefan–Boltzmann law writes total radiated power as P equals epsilon times sigma times A times T to the fourth. When power is measured or specified and temperature is the unknown, algebra inverts that product: T equals the fourth root of P over the product of epsilon, sigma, and A.
Area A is the radiating surface area, epsilon is emissivity (one for a perfect black body, typically between zero and one for real materials), and sigma is a universal constant. Absolute temperature is reported in kelvin because the law is written on the thermodynamic temperature scale. Zero power with positive area and emissivity is valid and yields zero kelvin.
Where temperature-from-power checks fit in study, design, and software
Students meet the inverted Stefan–Boltzmann law when a lab measures radiated power from a heated plate and asks for equivalent black-body temperature, when a contest problem gives watts, area, and emissivity and asks for kelvin without a full radiative-transfer simulation, or when a homework set supplies power and area for a gray body and expects T to four significant figures. Teachers can pre-generate answer keys for modest power and area 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 recover surface temperature from a known radiative load before a process sketch, or to satisfy programming-challenge statements that demand kelvin rather than a hand-waved estimate. Process and product first-order checks use the inverted law when only radiated power, a known emissivity, and surface area are available before view factors, multi-surface exchange, or convection are modeled.
What you can do with it
Homework and lab checks
Recover absolute temperature T = (P/(εσA))^(1/4) from measured or assigned radiated power, area, and emissivity before grading radiation worksheets or black-body labs.
Process and tutoring software
Normalize user-entered power, ε, and area into absolute temperature for heat-loss sketches, tutoring UIs, and interactive lessons.
FAQ
What does the temperature from radiated power calculator cost?
It is free in your browser on this page.
How is temperature defined from radiated power?
By the inverted Stefan–Boltzmann law, T = (P / (ε · σ · A))^(1/4), where P is radiated power in watts, A is surface area, ε is emissivity, and σ = 5.670374419e-8 W/(m²·K⁴). With SI units, T is in kelvin.
What happens if power is negative?
Radiated power cannot be negative; the inverted Stefan–Boltzmann law is undefined for negative power in this calculator.
Why must area and emissivity be strictly positive?
The inverted formula divides by the product ε·σ·A.
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/optics/temperature-from-radiated-power \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"power":459.3,"area":1,"emissivity":1}'const res = await fetch("https://api.kit.forhosting.com/optics/temperature-from-radiated-power", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"power": 459.3,
"area": 1,
"emissivity": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/temperature-from-radiated-power",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"power": 459.3,
"area": 1,
"emissivity": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/temperature-from-radiated-power", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"power":459.3,"area":1,"emissivity":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"power":459.3,"area":1,"emissivity":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/temperature-from-radiated-power", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"power": 459.3,
"area": 1,
"emissivity": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.temperature_from_radiated_power",
"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.
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 inverted Stefan–Boltzmann law means and what the response contains
This calculator returns the normalized inputs power, area, and emissivity, the fixed sigma it used, the computed temperature, the precision used, a method tag inverted_stefan_boltzmann, a short formula reminder, and a unit hint K for SI inputs. There is no automatic material lookup, no surrounding-environment term such as net exchange with T_env to the fourth, and no Monte Carlo sampling: the engine either evaluates the fourth-root expression inside the supported numeric range or rejects the input with invalid_input. Rejected cases include a missing power, area, or emissivity, non-finite values, magnitudes above the documented ceilings, negative power, and any area or emissivity that is not strictly positive, because the inverted law divides by those quantities.
How power, area, and emissivity are read and how T stays reproducible
Zero power is allowed and returns zero kelvin when area and emissivity are positive. Area must be finite and strictly positive: zero or negative area fails because the inverted law divides by A. 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 non-positive 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 radiated power is power, but aliases P, radiated_power, and watts are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. Power 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 cannot be negative in this calculator, so a bad cell does not become a silent nonsense kelvin result. The preferred field for surface area is area, with aliases A, surface_area, and surface. The preferred field for emissivity is emissivity, with aliases e, eps, and epsilon so textbook and engineering forms map cleanly. 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 inverted fourth-power formula yourself.
Where temperature-from-power checks fit in study, design, and software
Product and education tools use the endpoint when a learner types power, epsilon, and area and the UI must show absolute temperature. 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 the forward Stefan–Boltzmann power calculator when you need P from T, with other thermo or optics 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 kelvin values for unit tests, thermal demos, and CI assertions on the inverted Stefan–Boltzmann law.