⊙ 圆的方程计算器（标准式、一般式互化）Equation of a Circle Calculator (Standard & General Form)
由圆心和半径、圆心和圆上一点、直径的两个端点或三个点写出圆的标准方程 (x − h)² + (y − k)² = r² 和一般式 x² + y² + Dx + Ey + F = 0；也可以把一般式化回圆心和半径。同时给出面积和周长。本地处理。Write the standard equation (x − h)² + (y − k)² = r² and the general form x² + y² + Dx + Ey + F = 0 of a circle from its centre and radius, its centre and a point, the two ends of a diameter or three points, or turn a general form back into the centre and radius. The area and circumference are shown too. Processed locally.
使用说明与常见问题How-to & FAQ
「圆的方程计算器（标准式、一般式互化）」是什么？What is the Equation of a Circle Calculator (Standard & General Form) tool?
由圆心和半径、圆心和圆上一点、直径的两个端点或三个点写出圆的标准方程 (x − h)² + (y − k)² = r² 和一般式 x² + y² + Dx + Ey + F = 0；也可以把一般式化回圆心和半径。同时给出面积和周长。本地处理。Write the standard equation (x − h)² + (y − k)² = r² and the general form x² + y² + Dx + Ey + F = 0 of a circle from its centre and radius, its centre and a point, the two ends of a diameter or three points, or turn a general form back into the centre and radius. The area and circumference are shown too. Processed locally.
如何使用「圆的方程计算器（标准式、一般式互化）」？How do I use the Equation of a Circle Calculator (Standard & General Form) tool?
在「圆的方程计算器（标准式、一般式互化）」中，填写「已知」、「圆心 (h, k)」、「半径 r」、「点 1」、「点 2」等参数，结果会随输入实时计算并显示在下方。In the Equation of a Circle Calculator (Standard & General Form): Fill in "Given", "Centre (h, k)", "Radius r", "Point 1", "Point 2"; results are recalculated instantly as you type.
标准式和一般式怎么互化？How do the standard and general forms convert?
展开 (x − h)² + (y − k)² = r² 得到 x² + y² − 2hx − 2ky + (h² + k² − r²) = 0，所以 D = −2h、E = −2k、F = h² + k² − r²。反过来 h = −D/2、k = −E/2、r² = h² + k² − F。Expanding (x − h)² + (y − k)² = r² gives x² + y² − 2hx − 2ky + (h² + k² − r²) = 0, so D = −2h, E = −2k and F = h² + k² − r². Going back, h = −D/2, k = −E/2 and r² = h² + k² − F.
一般式一定是圆吗？Is every general form a circle?
不一定。算出的 r² = h² + k² − F 大于 0 才是圆；等于 0 只是一个点；小于 0 没有实数点。页面会提示这两种情况。No. It is a circle only if r² = h² + k² − F is positive; zero gives a single point and a negative value gives no real points. The page tells you in both cases.
三个点怎么确定圆？How do three points fix a circle?
不在一条直线上的三个点恰好确定一个圆，圆心是三边垂直平分线的交点（外心）。三点共线时没有这样的圆。Any three points not on one line lie on exactly one circle, centred where the perpendicular bisectors meet (the circumcentre). Collinear points have no such circle.
能举个计算实例吗？Can you show a worked example?
例如在「圆的方程计算器（标准式、一般式互化）」中填 已知 = 圆心和半径，圆心 (h, k) = 0, 0，半径 r = 5，结果区显示：标准式：x² + y² = 25；一般式：x² + y² − 25 = 0；圆心 (h, k)：(0, 0)；半径 r：5；面积 πr²：78.539816；周长 2πr：31.415927For example, entering Given = Centre and radius, Centre (h, k) = 0, 0, Radius r = 5 in the Equation of a Circle Calculator (Standard & General Form) gives: Standard form: x² + y² = 25; General form: x² + y² − 25 = 0; Centre (h, k): (0, 0); Radius r: 5; Area πr²: 78.539816; Circumference 2πr: 31.415927
哪些输入会被检查？Which inputs are validated?
「圆的方程计算器（标准式、一般式互化）」会检查这些输入——「半径 r」：所选方式用到时必须填写，须大于 0；「D」：所选方式用到时必须填写；「E」、「F」：可留空；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 Equation of a Circle Calculator (Standard & General Form) checks its inputs — "Radius r": required when the chosen option uses it, must be greater than 0; "D": required when the chosen option uses it; "E", "F": optional; 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?
「圆的方程计算器（标准式、一般式互化）」的设置：「已知」可选 圆心和半径、圆心和圆上一点、直径的两个端点、圆上三点、一般式 D、E、F（默认 圆心和半径）；「圆心 (h, k)」默认 2, -3；「半径 r」默认 5；「点 1」默认 5, 1；「点 2」默认 -1, -7；「点 3」默认 7, -3；「D」默认 -4；「E」默认 6；「F」默认 -12。每次生成结果都按当前设置处理。Settings in the Equation of a Circle Calculator (Standard & General Form) — "Given": Centre and radius, Centre and a point, Ends of a diameter, Three points, General form D, E, F (default Centre and radius); "Centre (h, k)" default 2, -3; "Radius r" default 5; "Point 1" default 5, 1; "Point 2" default -1, -7; "Point 3" default 7, -3; "D" default -4; "E" default 6; "F" default -12. The result always uses the current settings.
计算精度如何？How precise are the results?
「圆的方程计算器（标准式、一般式互化）」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 Equation of a Circle Calculator (Standard & General Form) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「圆的方程计算器（标准式、一般式互化）」免费吗？需要注册吗？Is the Equation of a Circle Calculator (Standard & General Form) free? Do I need an account?
「圆的方程计算器（标准式、一般式互化）」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 Equation of a Circle Calculator (Standard & General Form) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.