📍 重心计算器（三角形 / 多边形）Centroid Calculator (triangle or polygon)

使用说明与常见问题How-to & FAQ

「重心计算器（三角形 / 多边形）」是什么？What is the Centroid Calculator (triangle or polygon) tool?

求三角形（三个顶点）或任意简单多边形（按顺序列出顶点，用鞋带公式）的面积重心和面积，三点共线 / 面积为 0 时给出提示。本地处理。Area centroid and area of a triangle (three vertices) or any simple polygon (vertices in order, shoelace formula), with a warning when the points are collinear / the area is 0. Processed locally.

如何使用「重心计算器（三角形 / 多边形）」？How do I use the Centroid Calculator (triangle or polygon) tool?

在「重心计算器（三角形 / 多边形）」中，填写「图形」、「x1」、「y1」、「x2」、「y2」等参数，结果会随输入实时计算并显示在下方。In the Centroid Calculator (triangle or polygon): Fill in "Shape", "x1", "y1", "x2", "y2"; results are recalculated instantly as you type.

三角形的重心怎么算？How is a triangle's centroid found?

三个顶点坐标的平均值：x =（x1 + x2 + x3）÷ 3，y 同理，也是三条中线的交点。Average the three vertices: x = (x1 + x2 + x3) ÷ 3, and the same for y; it is where the three medians meet.

多边形用什么公式？Which formula is used for polygons?

鞋带公式：A = ½ Σ (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)，Cx = Σ (xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) ÷ (6A)，Cy 同理。顶点要沿边界按顺序输入（顺时针或逆时针均可），多边形不能自交。The shoelace formula: A = ½ Σ (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ), Cx = Σ (xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) ÷ (6A), and likewise Cy. List the vertices in order around the boundary (either direction); the polygon must not cross itself.

为什么提示共线？Why the collinear warning?

三点在一条直线上（或多边形面积为 0）时没有面积，面积重心无定义；页面改为显示顶点平均值供参考。When the points lie on one line (or the polygon has zero area) there is no area, so the area centroid is undefined; the page shows the average of the vertices instead.

能举个计算实例吗？Can you show a worked example?

例如在「重心计算器（三角形 / 多边形）」中填 图形 = 三角形，x1 = 0，y1 = 0，x2 = 6，y2 = 0，x3 = 0，结果区显示：重心 x：2；重心 y：2；面积：18For example, entering Shape = Triangle, x1 = 0, y1 = 0, x2 = 6, y2 = 0, x3 = 0 in the Centroid Calculator (triangle or polygon) gives: Centroid x: 2; Centroid y: 2; Area: 18

哪些输入会被检查？Which inputs are validated?

「重心计算器（三角形 / 多边形）」会检查这些输入——「x1」、「y1」、「x2」、「y2」、「x3」、「y3」：必须填写；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 Centroid Calculator (triangle or polygon) checks its inputs — "x1", "y1", "x2", "y2", "x3", "y3": required; 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?

「重心计算器（三角形 / 多边形）」的设置：「图形」可选 三角形、多边形（默认 三角形）；「x1」默认 0；「y1」默认 0；「x2」默认 4；「y2」默认 0；「x3」默认 2；「y3」默认 6。每次生成结果都按当前设置处理。Settings in the Centroid Calculator (triangle or polygon) — "Shape": Triangle, Polygon (default Triangle); "x1" default 0; "y1" default 0; "x2" default 4; "y2" default 0; "x3" default 2; "y3" default 6. The result always uses the current settings.

计算精度如何？How precise are the results?

「重心计算器（三角形 / 多边形）」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 Centroid Calculator (triangle or polygon) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「重心计算器（三角形 / 多边形）」免费吗？需要注册吗？Is the Centroid Calculator (triangle or polygon) free? Do I need an account?

「重心计算器（三角形 / 多边形）」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 Centroid Calculator (triangle or polygon) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.