⬡ 正多边形计算器（已知任意一个量求全部）Regular Polygon Calculator (Any One Measurement Gives the Rest)

使用说明与常见问题How-to & FAQ

「正多边形计算器（已知任意一个量求全部）」是什么？What is the Regular Polygon Calculator (Any One Measurement Gives the Rest) tool?

输入边数和任意一个已知量（边长、外接圆半径、内切圆半径/边心距、周长或面积），求正多边形的其余所有量：边长、周长、面积、外接圆半径、边心距、内角、外角、中心角和对角线条数。本地处理。Enter the number of sides and any one known measurement (side, circumradius, inradius or apothem, perimeter or area) to get every other property of the regular polygon: side, perimeter, area, circumradius, apothem, interior, exterior and central angles, and the number of diagonals. Processed locally.

如何使用「正多边形计算器（已知任意一个量求全部）」？How do I use the Regular Polygon Calculator (Any One Measurement Gives the Rest) tool?

在「正多边形计算器（已知任意一个量求全部）」中，填写「边数」、「已知」、「数值」等参数，结果会随输入实时计算并显示在下方。In the Regular Polygon Calculator (Any One Measurement Gives the Rest): Fill in "Sides", "Known", "Value"; results are recalculated instantly as you type.

面积公式是什么？What is the area formula?

正 n 边形面积 = n × s² ÷ (4 × tan(π/n))，也等于 ½ × 周长 × 边心距。例如边长 2 的正六边形面积约 10.392。Area = n × s² / (4 × tan(π/n)), which is also ½ × perimeter × apothem. A regular hexagon with side 2 has area about 10.392.

半径和边长怎么换算？How do radius and side relate?

外接圆半径 R = s ÷ (2 sin(π/n))，边心距 r = s ÷ (2 tan(π/n))。正六边形的边长恰好等于外接圆半径。Circumradius R = s / (2 sin(π/n)) and apothem r = s / (2 tan(π/n)). In a regular hexagon the side equals the circumradius.

内角怎么算？How are the angles found?

内角 = (n − 2) × 180° ÷ n，外角 = 360° ÷ n，二者之和为 180°。对角线条数 = n(n − 3) ÷ 2。Interior angle = (n − 2) × 180° / n and exterior angle = 360° / n; together they make 180°. The number of diagonals is n(n − 3) / 2.

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

例如在「正多边形计算器（已知任意一个量求全部）」中填 边数 = 6，已知 = 边长，数值 = 2，结果区显示：名称：正六边形；边长：2；周长：12；面积：10.39230485；外接圆半径 R：2；边心距 r：1.73205081；内角：120°；外角 / 中心角：60°；内角和：720°；对角线条数：9For example, entering Sides = 6, Known = Side, Value = 2 in the Regular Polygon Calculator (Any One Measurement Gives the Rest) gives: Name: regular hexagon; Side: 2; Perimeter: 12; Area: 10.39230485; Circumradius R: 2; Apothem r: 1.73205081; Interior angle: 120°; Exterior / central angle: 60°; Sum of interior angles: 720°; Diagonals: 9

哪些输入会被检查？Which inputs are validated?

「正多边形计算器（已知任意一个量求全部）」会检查这些输入——「边数」：必须填写，须为 3–100,000 的整数；「数值」：必须填写，须大于 0；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 Regular Polygon Calculator (Any One Measurement Gives the Rest) checks its inputs — "Sides": required, must be a whole number from 3 to 100,000; "Value": required, must be greater than 0; 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?

「正多边形计算器（已知任意一个量求全部）」的设置：「边数」默认 6；「已知」可选 边长、外接圆半径、边心距（内切圆半径）、周长、面积（默认 边长）；「数值」默认 2。每次生成结果都按当前设置处理。Settings in the Regular Polygon Calculator (Any One Measurement Gives the Rest) — "Sides" default 6; "Known": Side, Circumradius, Apothem (inradius), Perimeter, Area (default Side); "Value" default 2. The result always uses the current settings.

计算精度如何？How precise are the results?

「正多边形计算器（已知任意一个量求全部）」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 Regular Polygon Calculator (Any One Measurement Gives the Rest) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「正多边形计算器（已知任意一个量求全部）」免费吗？需要注册吗？Is the Regular Polygon Calculator (Any One Measurement Gives the Rest) free? Do I need an account?

「正多边形计算器（已知任意一个量求全部）」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 Regular Polygon Calculator (Any One Measurement Gives the Rest) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.