Hypocycloid Calculator
Free online hypocycloid 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：内摆线在线计算器
内摆线是一个圆沿定圆内侧滚动时圆上定点的轨迹：x = (R−r)cos t + r cos((R−r)t/r)。尖点数 = R/r。当 R = 2r 时奇迹发生——轨迹退化为一条直径（卡丹圆盘原理：小圆半径是大圆一半时，圆上任意点走直线，早期圆规车床靠它切出直线滑槽）。
R/r 为整数时周长 L = 8(R−r)、面积 A = π(R−r)(R−2r)。星形线（R=4r）即 x^(2/3)+y^(2/3) = R^(2/3)，面积 π(R−r)(R−2r) = 6π（R=4、r=1），与经典公式 3πa²/8（a = R）一致；三尖内摆线（R=3r）面积 2π。本工具以 100 万点 shoelace 全部验证。
参考值
星形线 R=4、r=1：L = 24、A = 6π = 18.849555921539
三尖内摆线 R=3、r=1：L = 16、A = 2π = 6.283185307180；R = 2r：退化为直径线段
How to use the Hypocycloid 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 Hypocycloid 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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