Gabriel Horn Calculator
Free online gabriel horn 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：加布里埃尔号角在线计算器
原理：曲线 y = 1/x（x ≥ 1）绕 x 轴旋转得喇叭形旋转体。体积 V = π∫₁ᵇ dx/x² = π(1 − 1/b)，当 b → ∞ 时收敛于 π；而侧面积 S = 2π∫₁ᵇ (1/x)√(1 + 1/x⁴) dx ≥ 2π∫₁ᵇ dx/x = 2π ln b 发散于无穷。
步骤：输入右端点 b，得到有限区间上的体积与侧面积；观察 b 增大时体积逼近 π 而面积持续增长。
示例：b = 10 时 V ≈ 2.827433388231；b → ∞ 时 V = 3.141592653590（恰好 π），S 却无界。
注意事项：这就是「油漆悖论」——号角内部装不满一升油漆，其内壁却刷不完一桶油漆；它提醒我们「体积收敛」与「表面积收敛」是两回事。
参考值：V(∞) = 3.141592653590；S(1→10) = 15.177449874820（数值）。
How to use the Gabriel Horn 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 Gabriel Horn 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 2026-10-01
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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