🍩 圆环体（环面）体积计算器Torus Volume Calculator
输入圆环的主半径 R（中心到管中心）和管半径 r，按 V = 2π²Rr² 计算体积、按 S = 4π²Rr 计算表面积；也可以改用内半径和外半径输入。Enter the major radius R (centre to tube centre) and tube radius r of a torus to get its volume V = 2π²Rr² and surface area S = 4π²Rr — or enter the inner and outer radii instead.
使用说明与常见问题How-to & FAQ
「圆环体（环面）体积计算器」是什么？What is the Torus Volume Calculator tool?
输入圆环的主半径 R（中心到管中心）和管半径 r，按 V = 2π²Rr² 计算体积、按 S = 4π²Rr 计算表面积；也可以改用内半径和外半径输入。Enter the major radius R (centre to tube centre) and tube radius r of a torus to get its volume V = 2π²Rr² and surface area S = 4π²Rr — or enter the inner and outer radii instead.
如何使用「圆环体（环面）体积计算器」？How do I use the Torus Volume Calculator tool?
在「圆环体（环面）体积计算器」中，填写「输入方式」、「主半径 R」、「管半径 r」、「内半径」、「外半径」等参数，结果会随输入实时计算并显示在下方。In the Torus Volume Calculator: Fill in "Input", "Major radius R", "Tube radius r", "Inner radius", "Outer radius"; results are recalculated instantly as you type.
圆环体体积公式怎么来的？Where does the torus volume formula come from?
由帕普斯–古尔丁定理：把半径为 r 的圆绕距离圆心 R 的轴旋转一周，体积 = 圆面积 πr² × 圆心走过的路程 2πR = 2π²Rr²；表面积 = 圆周长 2πr × 2πR = 4π²Rr。By Pappus's centroid theorem: sweeping a circle of radius r around an axis R from its centre gives volume = circle area πr² × the path of its centre 2πR = 2π²Rr², and surface area = circumference 2πr × 2πR = 4π²Rr.
内半径和外半径怎么换算？How do inner and outer radii convert?
主半径 R = (内半径 + 外半径) ÷ 2，管半径 r = (外半径 − 内半径) ÷ 2。例如甜甜圈内半径 5、外半径 10，则 R = 7.5、r = 2.5。Major radius R = (inner + outer) ÷ 2 and tube radius r = (outer − inner) ÷ 2. A doughnut with inner radius 5 and outer radius 10 has R = 7.5 and r = 2.5.
为什么 r 不能大于 R？Why can't r exceed R?
r > R 时管子会穿过中心轴、自相交（纺锤环面），重叠部分会被公式重复计算，结果不再是实际体积，所以不予计算。When r > R the tube passes through the axis and overlaps itself (a spindle torus); the formula would count the overlap twice, so it no longer gives the real volume and is not calculated.
计算精度如何？How precise are the results?
「圆环体（环面）体积计算器」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Torus Volume Calculator uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「圆环体（环面）体积计算器」免费吗？需要注册吗？Is the Torus Volume Calculator free? Do I need an account?
「圆环体（环面）体积计算器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Torus Volume Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.