Thin prism deviation calculator
A thin prism of index 1.5 with a 4° apex turns a ray by 2°. Enter the refractive index and the apex angle in degrees to get the deviation and the index-minus-one factor for your lab notebook. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Refractive index
- 1.5
- Apex angle
- 4
- N minus one
- 0.5 degrees
- Deviation
- 2
- Approximation
- thin prism, small angle
A thin prism bends a ray by a nearly constant angle when the apex is small and angles stay in the paraxial regime. The textbook formula is simple: the deviation angle delta equals the refractive index minus one, multiplied by the apex angle. This calculator takes the prism index n and the apex angle A in degrees, multiplies (n − 1) by A, and returns the deviation in the same degree unit, along with the intermediate (n − 1) factor for lab notebooks. No network calls, no random seeds, and no wall-clock timestamps enter the math.
How thin-prism deviation is derived and when it applies
Start with a triangular prism of refractive index n greater than one and apex angle A at the refracting edge. A ray that enters the first face and exits the second is bent twice by Snell's law. The total turn of the ray relative to the undeviated direction is the deviation angle delta.
When both the apex and the angles of incidence and emergence are small enough that the small-angle approximation sine theta approximately equals theta in radians holds, the two refractions collapse to a single compact result: delta approximately equals (n minus one) times A. In that limit the deviation no longer depends on the exact incidence angle, which is why lab scripts often quote only n and A.
The formula is linear in A, so if you express A in degrees the returned deviation is also in degrees; if you used radians throughout, the same product would be in radians. This capability works in degrees, matching school and college problem sets. It does not solve the full prism equation for minimum deviation with large apex angles, does not invert measured deviation to recover n, and does not ray-trace curved faces or multilayer coatings.
Both inputs must be finite; n must be strictly greater than one; A must be positive and less than one hundred eighty degrees so the apex remains a meaningful prism geometry.
Where thin-prism deviation calculators show up in real work
Secondary-school and college optics courses assign endless thin-prism problems that ask only for delta given n and A; a single trusted oracle that returns degrees plus the intermediate (n − 1) speeds grading and demo setup. Introductory spectrometer labs estimate how much a dispersing wedge of known apex will steer a monochromatic beam before students move to full minimum-deviation measurements on thicker prisms. Optical-engineering notebooks use the linear rule as a first-order check when specifying small wedge prisms for beam steering, alignment, or low-dispersion windows.
If you later need Snell's law at a single flat interface, critical-angle total internal reflection, Brewster polarization angles, thin-lens image conjugates, or full prism minimum-deviation inversion for large apex angles, treat those as separate search intents and landings rather than overloading this thin-prism small-angle deviation page.
What you can do with it
Homework and lab answer keys
Compute thin-prism deviation δ = (n − 1)A from textbook refractive index and apex angle so teachers and auto-graders share one ground truth in degrees.
Spectrometer and dispersing-wedge demos
Estimate how far a small apex prism steers a monochromatic beam before students measure minimum deviation on thicker prisms.
First-order wedge prism design checks
Quickly size small steering or alignment wedges from glass index and apex when a full ray-trace is not yet justified.
FAQ
What formula does this thin prism deviation calculator use?
δ = (n − 1) · A under the thin-prism small-angle approximation. n is the refractive index, A is the apex angle in degrees, and δ is returned in degrees.
Does deviation depend on the angle of incidence?
Not in this approximation. For small apex and small incidence, textbooks show that δ is independent of incidence and depends only on n and A. Large-angle minimum-deviation formulas are a different tool.
What units should I use for the apex angle?
Degrees. Because δ = (n − 1)A is linear, the deviation is reported in the same unit as A; this capability standardizes on degrees for school and lab problems.
What refractive index values are accepted?
Any finite n strictly greater than 1. Vacuum or air near 1 is rejected because a prism requires denser material; crown glass is about 1.52 and flint about 1.62.
What does it cost?
The browser tool is free and runs locally.
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/thin-prism-deviation \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"refractive_index":1.5,"apex_angle":4}'const res = await fetch("https://api.kit.forhosting.com/optics/thin-prism-deviation", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"refractive_index": 1.5,
"apex_angle": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/thin-prism-deviation",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"refractive_index": 1.5,
"apex_angle": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/thin-prism-deviation", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"refractive_index":1.5,"apex_angle":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"refractive_index":1.5,"apex_angle":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/thin-prism-deviation", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"refractive_index": 1.5,
"apex_angle": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.thin_prism_deviation",
"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 you send and what you get back
Both fields are required. Classic checks include crown glass near n equals one point five with a four-degree apex giving two degrees of deviation, and flint glass near n equals one point six two with a five-degree apex giving three point one degrees. Wire the free browser form for homework and spectrometer demos, then call the identical contract from scripts when you need bulk answer keys or automated worksheet grading. Send two finite numbers under the fields refractive_index and apex_angle. refractive_index is the absolute index n of the prism glass or plastic relative to vacuum; apex_angle is A in degrees. Aliases such as n, index, or n_prism for the index and A, alpha, apex, or angle_apex for the apex are accepted so scripts written against common textbook notation do not need a rename layer. Missing fields, non-numeric strings, NaN, Infinity, n less than or equal to one, non-positive apex, and apex of one hundred eighty degrees or more fail with invalid_input and are not charged. On success the response echoes cleaned refractive_index and apex_angle, returns n_minus_one as the intermediate (n − 1), deviation as the product in degrees, plus formula set to the thin-prism expression, unit set to degrees, and approximation noting the thin prism small-angle regime. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where thin-prism deviation calculators show up in real work
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.