Focal length from lens radii
Turn a glass index and two surface radii into a focal length: an index of 1.5 with radii of 0.1 m and -0.1 m gives 0.1 m and 10 diopters, a converging lens. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- N
- 1.5
- R1
- 0.1
- R2
- -0.1
- R1 m
- 0.1
- R2 m
- -0.1
- Optical power dpt
- 10
- Focal length m
- 0.1
This calculator finds the thin-lens focal length from the material refractive index and the two surface radii of curvature. You receive focal length in meters, optical power in diopters, and a converging, diverging, or afocal label. Use it for optics homework, lab design, singlet sizing for cameras and collimators, and any automation pipeline that needs a transparent closed-form result without spreadsheets or proprietary design packages. Index equal to one is rejected because the element would match the surrounding medium and produce no optical power.
Lensmaker equation assumptions and when the result is valid
The thin-lens lensmaker formula is the first-order model taught throughout introductory optics courses and used as a baseline in early product design. It assumes that the lens center thickness is negligible compared with the surface radii and with object and image distances, that both faces are spherical or planar, and that the surrounding medium is air with refractive index essentially one. Under those conditions the two surface powers add, and the combined optical power is (n minus one) times the difference of the surface curvatures.
Thick lenses need the full thick-lens expression with center thickness and possibly different object- and image-space indices; aspheres, gradient-index glass, cemented doublets, and multi-element objectives need ray tracing or manufacturer prescription data. This capability deliberately returns the ideal thin-lens closed form so you can check textbook answers, seed a concept design before detailed modeling, or document a baseline power in logs and reports.
Radii follow the Cartesian sign convention with light traveling left to right: a radius is positive when the center of curvature lies to the right of the surface. Planar faces are accepted as radius zero, contributing zero curvature and matching the usual student-calculator convention for plano-convex and plano-concave elements.
Where teams use focal length from radii in practice
Physics and engineering courses use this calculator to verify homework and lab prep: students pick crown glass near n equals 1.5, choose symmetric biconvex radii, and confirm that f matches the expected value before grinding or measuring on an optical bench. Product teams sizing simple magnifiers, LED collimators, or phone-camera singlet concepts need a transparent power estimate before they open a full optical design package.
Instrumentation groups compare measured focal lengths against the thin-lens prediction to flag wrong glass type, inverted surface order, or a setup that is no longer thin.
What you can do with it
Quick singlet power for product concepts
Estimate converging or diverging power for a glass or plastic singlet during early camera, magnifier, or collimator sizing before detailed design.
FAQ
Why is refractive index equal to one rejected?
If n equals one, the lens matches the surrounding medium and the formula gives zero optical power for any radii.
Does this handle thick lenses or multi-element systems?
No. This capability returns the standard thin-lens closed form only. Thick lenses need center thickness and a fuller formula; multi-element systems need ray tracing or manufacturer prescriptions.
How do I enter a flat lens surface?
Enter zero as the radius of the flat face. A plane surface adds no curvature, which is the usual convention for plano-convex and plano-concave lenses in student calculators.
What sign should each lens radius have?
With light traveling left to right, a radius is positive when its center of curvature lies to the right of the surface. A symmetric biconvex lens therefore has a positive first radius and a negative second one.
What focal length does a biconvex lens with index 1.5 and radii of 0.1 m have?
With radii of 0.1 m and -0.1 m, the focal length is 0.1 m and the optical power is 10 diopters. The result labels the lens as converging.
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/focal-length-from-radii \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":1.5,"r1":0.1,"r2":-0.1}'const res = await fetch("https://api.kit.forhosting.com/optics/focal-length-from-radii", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 1.5,
"r1": 0.1,
"r2": -0.1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/focal-length-from-radii",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 1.5,
"r1": 0.1,
"r2": -0.1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/focal-length-from-radii", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":1.5,"r1":0.1,"r2":-0.1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":1.5,"r1":0.1,"r2":-0.1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/focal-length-from-radii", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": 1.5,
"r1": 0.1,
"r2": -0.1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.focal_length_from_radii",
"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
Inputs, outputs, units, and validation rules
All three must be finite. Either radius may be zero to mean a flat surface; non-zero radii may be positive or negative according to the sign convention. Convert millimeters to meters before calling if your drawings use mm; convert diopters to focal length with f equals one over P when you only know power. The index must be strictly positive and must not equal one—when n equals one the formula predicts zero power for any radii, which is unphysical for a focusing element and is rejected with invalid_input so clients fail fast. The response echoes n, r1_m, and r2_m, then optical_power_dpt (diopters when lengths are in meters), focal_length_m (meters, or null when power is exactly zero for an afocal combination), lens_type (converging, diverging, or afocal), and the formula string for audit logs. Missing fields, non-numeric strings, non-positive index, or index exactly one fail with invalid_input and are not charged. Pricing for a successful analytic call is a flat $0.002 per item.
Where teams use focal length from radii in practice
Because each successful call is a single analytic item at $0.002, scripts can sweep families of radii or index tables without a subscription or model cost. Prefer this endpoint when you want a stable, reviewable closed form—not a thick-lens solver, not multi-element ray tracing, and not wavelength-dependent dispersion, which require additional thickness, spacing, and glass catalog inputs beyond this contract.
Integration use cases
Homework and lab expected focal length
Compute the ideal thin-lens f from n, R1, and R2 before students measure focal length on an optical bench and report percent error.
Parameter sweeps in design scripts
Batch-call the API across radius pairs and index tables to build focal-length charts without spreadsheets or opaque desktop calculators.
Technical questions
What formula does this focal length from radii calculator use?
It uses the thin-lens lensmaker form in air: 1/f = (n − 1)(1/R1 − 1/R2). Outputs include focal length in meters, optical power in diopters, and a converging/diverging/afocal label.
What sign convention do R1 and R2 use?
Cartesian convention with light left to right: a radius is positive when the center of curvature is to the right of the surface. A typical biconvex lens has R1 > 0 and R2 < 0.
How much does each API call cost?
$0.002 per successful request. The browser tool above runs free locally; the API draws from your prepaid KIT balance only when the task succeeds.