⊙ 圆的切线长计算器Tangent Length to a Circle Calculator
圆外一点到切点的切线长 L = √(d² − r²)，d 为该点到圆心的距离，r 为半径（须 d > r）。本地处理。Tangent length from an external point L = √(d² − r²), with d = distance to centre and r = radius (need d > r). Processed locally.
使用说明与常见问题How-to & FAQ
「圆的切线长计算器」是什么？What is the Tangent Length to a Circle Calculator tool?
圆外一点到切点的切线长 L = √(d² − r²)，d 为该点到圆心的距离，r 为半径（须 d > r）。本地处理。Tangent length from an external point L = √(d² − r²), with d = distance to centre and r = radius (need d > r). Processed locally.
如何使用「圆的切线长计算器」？How do I use the Tangent Length to a Circle Calculator tool?
在「圆的切线长计算器」中，填写「圆心距」、「半径」等参数，结果会随输入实时计算并显示在下方。In the Tangent Length to a Circle Calculator: Fill in "Distance to centre", "Radius"; results are recalculated instantly as you type.
为什么两条切线一样长？Why are the two tangents equal?
圆心到切点的半径垂直于切线，两个直角三角形全等，故切线长相等。The radius to a point of tangency is perpendicular to the tangent; the two right triangles are congruent.
点在圆上或圆内呢？What if the point is on or inside the circle?
圆上切线长为 0；圆内没有实切线。本页要求圆心距大于半径。On the circle the length is 0; inside there is no real tangent. This page requires distance > radius.
能举个计算实例吗？Can you show a worked example?
例如在「圆的切线长计算器」中填 圆心距 = 13，半径 = 5，结果区显示：切线长：12For example, entering Distance to centre = 13, Radius = 5 in the Tangent Length to a Circle Calculator gives: Tangent length: 12
哪些输入会被检查？Which inputs are validated?
「圆的切线长计算器」会检查这些输入——「圆心距」：必须填写，须大于 0且不大于 1,000,000,000,000,000；「半径」：必须填写，须大于 0且不大于 1,000,000,000,000,000；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Tangent Length to a Circle Calculator checks its inputs — "Distance to centre": required, must be more than 0 and at most 1,000,000,000,000,000; "Radius": required, must be more than 0 and at most 1,000,000,000,000,000; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.
有哪些选项可以调整？What options can I adjust?
「圆的切线长计算器」的设置：「圆心距」默认 13；「半径」默认 5。每次生成结果都按当前设置处理。Settings in the Tangent Length to a Circle Calculator — "Distance to centre" default 13; "Radius" default 5. The result always uses the current settings.
计算精度如何？How precise are the results?
「圆的切线长计算器」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 Tangent Length to a Circle Calculator uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「圆的切线长计算器」免费吗？需要注册吗？Is the Tangent Length to a Circle Calculator free? Do I need an account?
「圆的切线长计算器」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 Tangent Length to a Circle 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.