Epicycloid Calculator
Free online epicycloid calculator. Enter your values and get instant results with the formula, step-by-step working, a worked example and FAQs — everything runs locally in your browser on Calcton, no signup required.
中文版：外摆线在线计算器
外摆线是一个圆沿另一个圆外侧无滑动滚动时，圆上定点的轨迹：x = (R+r)cos t − r cos((R+r)t/r)。齿轮设计中摆线齿廓的抗磨与少齿差行星减速器的摆线轮都是它的工业化身；咖啡因分子命名外的「心形线与肾形线」也来自这个家族。
R/r 为整数 n 时曲线闭合、恰有 n 个尖点，且周长有优雅闭式 L = 8(R+r)——与 n 无关！面积 A = π(R+r)(R+2r)。本工具用 100 万点 shoelace 数值面积交叉验证：R=3、r=1 得 20π = 62.831853071796，心形线（R=r）6π、肾形线（R=2r）12π 全部吻合。
参考值
R=3、r=1：L = 32、A = 20π = 62.831853071796
心形线 R=r=1：L = 16、A = 6π = 18.849555921539；肾形线 R=2、r=1：L = 24、A = 12π = 37.699111843078
How to use the Epicycloid Calculator
- Fill in the inputs with your own numbers — every field shows what it expects.
- The result updates instantly as you type; no "Calculate" click needed.
- Read the result card for the main figure, the step-by-step formula, and practical notes.
Frequently asked questions
Is the Epicycloid Calculator free?
Yes — all math calculators on Calcton are free, with no sign-up required. Your inputs are processed in your browser and never sent to a server.
Is there a Chinese version of this tool?
Yes. Use the Chinese version link above — it is the same calculator with the interface in Simplified Chinese.
References
Formulas and benchmarks used by this calculator follow these published sources.
KevinContent LeadCalcton Editorial Team · Last updated 2026-09-29
Kevin maintains Calcton's formula verification and content: every calculator cites the standard it implements, worked examples are recomputed programmatically, and health benchmarks follow WHO and international guidelines.
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