Golden Section Calculator
Find the two golden-section division points within an interval [start, end] — the split where the whole is to the longer part as the longer part is to the shorter, φ ≈ 1.618. Used for golden-section search and layout/design ranges.
Input
Output
| Metric | Value |
|---|---|
| No data yet | |
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/golden-section-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"start": "0",
"end": "100",
"precision": "4"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `golden-section-calculator` tool (Golden Section Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/golden-section-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Golden Section Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
Also available with
Guides
The golden section splits an interval into two parts so that the whole is to the longer part what the longer part is to the shorter part — the same φ ≈ 1.618 proportion behind the golden ratio, but applied to a two-ended range [start, end] instead of a single length. That's the shape golden-section search (an optimization technique that narrows a search range without ever recomputing the ratio) and interval-based design layouts actually need: not "split this length," but "where inside this range do the golden-section points fall?"
How to use it
- Enter the interval start and interval end — any two numbers, in either order of magnitude, as long as end is greater than start.
- Choose a decimal precision. φ is irrational, so more decimals means less rounding error if you're feeding the result into another calculation.
The tool returns both golden-section points (one nearer the start, one nearer the end), the two segment lengths they create, and the golden ratio itself for a quick sanity check.
The math
For an interval of length L = end − start:
- Shorter segment =
L / φ² - Longer segment =
L / φ - Golden-section point nearer the start =
start + L / φ² - Golden-section point nearer the end =
end − L / φ²(equivalently,start + L / φ)
Both points are symmetric around the interval's center, and each divides the interval so the ratio of the whole segment to the longer part equals the ratio of the longer part to the shorter part — exactly φ.
How is this different from a golden-ratio split?
A golden-ratio calculation starts from one known length and derives the other two (see our Golden Ratio Calculator). This tool starts from a full interval — two endpoints — and finds where inside that range the golden-section points sit, which is what golden-section search and range-based layout problems actually hand you as a starting point.
Why two points instead of one?
Golden-section search evaluates a function at both interior points on each iteration and discards the sub-interval that can't contain the optimum, keeping the other point without recomputing it — that's the trick that makes it more efficient than an evenly-spaced grid search. Even outside optimization, having both points is useful any time you want to split a range symmetrically at the golden proportion — a composition guide, a print margin, or a UI panel divider.
Related tools
For splitting a single known length instead of an interval, use the Golden Ratio Calculator; for golden-ratio rectangle dimensions, see the Golden Rectangle Calculator.