Moon illuminated fraction calculator
Find how much of the Moon's face is lit from the Sun–Moon–observer angle: at a phase angle of 90 degrees exactly half the disk shines, a quarter moon. At 0 degrees it is full, at 180 new. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Phase angle
- 90
- Phase angle deg
- 90 deg
- Phase angle rad
- 1.5707963268
- Cos phase angle
- 0
- Illuminated fraction
- 0.5
The illuminated fraction of the Moon's disk is the share of the apparent lunar face that is lit by the Sun as seen from Earth. Observers, photographers, and educators often need that fraction from a known phase angle rather than from a calendar date. This moon illuminated fraction calculator takes the phase angle i of the Sun–Moon–observer geometry and returns k equals one plus cosine of i, all over two—the standard textbook relation used in lunar photometry and ephemeris notes. At zero degrees the disk is fully illuminated (full moon); at ninety degrees half the disk is lit (quarter); at one hundred eighty degrees the night side faces Earth (new moon).
What phase angle means and how the illuminated fraction is defined
In lunar geometry the phase angle i is the angle at the Moon between the directions to the Sun and to the observer. When the Sun, Moon, and Earth are nearly aligned with Earth in the middle, i is near zero and the near side is fully sunlit—full moon. When the Moon sits roughly ninety degrees from the Sun in the sky, i is about ninety degrees and half the disk is bright—first or last quarter.
When the Moon lies between Earth and the Sun, i approaches one hundred eighty degrees and the night hemisphere faces us—new moon. The classical illuminated fraction of the apparent disk is k equals (1 + cos i) / 2. That identity follows from the projected area of a spherical body: the terminator divides the sphere so the sunlit portion of the disk equals half of one plus the cosine of the phase angle.
The result is always between zero and one inclusive. This capability evaluates that formula only; it does not convert calendar dates into i, does not correct for libration or atmosphere, and does not model earthshine. If you already have i from an ephemeris, orbit propagator, or textbook problem, you can obtain k immediately.
If you need phase name or illumination from a civil date alone, use a moon-phase-from-date tool instead—this page targets the angle-to-fraction step that those pipelines still need as a pure function.
Where knowing the lit fraction helps
Photography planners and exposure notebooks convert known phase angles into a fill factor for histogram targets and HDR brackets. Classroom labs ask students to compute k from tabulated i values; an oracle that rejects out-of-range angles teaches the physical domain instead of silently clamping. Orbit and mission-analysis scripts already integrate solar and lunar vectors and need only k equals (1 + cos i) / 2 for disk illumination flags in reports.
Content sites that rank for moon illuminated fraction calculator benefit from a free browser path that matches the paid automation path on the same pure module. Failed validation does not consume balance. Keep requests small, pin the rounded fraction in tests, and you get reproducible disk illumination for education, media, and edge workers without standing up a full lunar ephemeris just to evaluate a cosine.
What you can do with it
Ephemeris post-processing
Turn a phase angle from any lunar ephemeris into the illuminated disk fraction k = (1 + cos i) / 2 for dashboards and reports.
Classroom and homework checks
Let students submit i in degrees or radians and verify k, percent, and out-of-range errors against textbook lunar geometry.
Photography and exposure planning
Convert known phase angles into fill fractions for histogram targets, HDR brackets, and lunar imaging checklists.
FAQ
Does this calculator use the date or the Moon's age?
No. It only evaluates k from a given phase angle. For phase from a civil date, use a separate moon-phase-from-date tool, then pass i here if you need the fraction formula alone.
What is the Moon's phase angle?
It is the angle measured at the Moon between the direction to the Sun and the direction to the observer. Small angles mean the side facing you is sunlit; large angles mean it is mostly dark.
How much of the Moon is lit at a phase angle of 90 degrees?
Exactly half, an illuminated fraction of 0.5 or 50 %. That is the first or last quarter, when the Moon sits roughly ninety degrees from the Sun in the sky.
Which phase angle gives a full moon and which a new moon?
A phase angle of 0 degrees gives a fully lit disk, the full moon. At 180 degrees the night side faces Earth and nothing is lit, the new moon.
Can I enter the angle in radians?
Yes. Degrees are used by default, but you can switch to radians. The angle must stay between 0 and 180 degrees, or between 0 and pi in radians.
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/astro/moon-illumination-fraction \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"phase_angle":90}'const res = await fetch("https://api.kit.forhosting.com/astro/moon-illumination-fraction", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"phase_angle": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/moon-illumination-fraction",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"phase_angle": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/moon-illumination-fraction", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"phase_angle":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"phase_angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/moon-illumination-fraction", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"phase_angle": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.moon_illumination_fraction",
"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
Angles outside the physical range are rejected with invalid_input so automation never treats an impossible geometry as a silent zero.
Input contract, units, range checks, and response shape
Missing, blank, or non-finite angles also fail. Numbers are rounded to twelve decimal places for stable goldens across runtimes. POST a JSON body with phase_angle, a finite number, and optionally unit set to deg (default) or rad. Aliases i, phase, angle, and phase_angle_deg are accepted for the angle field. When unit is deg, phase_angle must lie in the closed interval from zero to one hundred eighty; when unit is rad, it must lie in the closed interval from zero to π. Values below the lower bound or above the upper bound fail with invalid_input and a message that names the out-of-range condition—those failures are not billed. On success the body echoes phase_angle and phase_angle_deg in degrees, phase_angle_rad in radians, the unit used, cos_phase_angle, illuminated_fraction k in zero-to-one form, illuminated_percent as one hundred times k, a coarse phase_hint (full, gibbous, quarter, crescent, or new), the formula string, a substituted equation, and a short message. Example: phase_angle ninety with unit deg yields illuminated_fraction zero point five and illuminated_percent fifty, matching a geometric quarter moon. Pair the fraction with your own magnitude or exposure model if you need apparent brightness; this endpoint stops at the geometric disk fraction.
Where a phase-angle illuminated-fraction API earns its keep
Planetarium and outreach sites show live “how much of the Moon is lit” widgets once an ephemeris has produced i; calling a tiny parse API avoids re-implementing the cosine formula in every front-end language.
Technical questions
How do I compute the Moon's illuminated fraction from phase angle?
Send phase_angle i (0–180° by default, or 0–π rad with unit rad). The API returns k = (1 + cos i) / 2 as illuminated_fraction and illuminated_percent.
What is the allowed range for phase_angle?
With unit deg (default): 0 to 180 inclusive. With unit rad: 0 to π inclusive. Values outside that range fail with invalid_input and are not billed.
What does phase_angle 0°, 90°, and 180° mean?
0° → full moon (k = 1); 90° → geometric quarter (k = 0.5); 180° → new moon (k = 0). Intermediate values give gibbous or crescent fractions.
What does the moon illuminated fraction calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.