📚 Regular Polygons: Interior and Exterior Angles | 正多边形：内角与外角
A regular polygon is one where all sides and all angles are equal. A classic IGCSE question asks: ‘A regular polygon has an interior angle of 174°. How many sides does it have?’ This article unlocks the formulas and reasoning needed to solve such problems with confidence.
正多边形是所有边长和所有内角都相等的多边形。一道经典的 IGCSE 考题问：“一个正多边形的内角为 174°，它有多少条边？” 本文将从公式到推理，帮你轻松解决这类问题。
1. Key Definitions | 关键定义
An interior angle is an angle inside the polygon, formed by two adjacent sides. An exterior angle is the angle between a side and the extension of the adjacent side.
内角是多边形内部由两条相邻边形成的角。外角是一条边与相邻边延长线之间的角。
- For a regular polygon, all interior angles are equal.
- For a regular polygon, all exterior angles are equal.
- The sum of an interior angle and its adjacent exterior angle is always 180°.
- 正多边形的所有内角都相等。
- 正多边形的所有外角都相等。
- 一个内角与其相邻外角的和恒为 180°。
2. Sum of Interior Angles | 内角和
For any polygon with n sides, the sum of the interior angles is given by:
对于任何 n 边形，其内角和为：
Sum of interior angles = (n − 2) × 180°
This formula works for any straight-sided polygon, regular or irregular. You can see that a triangle (n = 3) gives 180°, a quadrilateral (n = 4) gives 360°, and so on.
这个公式适用于任何直线边多边形，无论正或不正。三角形 (n = 3) 得 180°，四边形 (n = 4) 得 360°，依此类推。
3. Exterior Angles and the 360° Rule | 外角与 360° 规律
At each vertex, the interior and exterior angles lie on a straight line, so they sum to 180°. For any polygon, if you walk around the perimeter, you turn through the exterior angles one at a time and make one full revolution. Therefore:
在每个顶点，内角和外角构成一条直线，因此它们之和为 180°。对于任何多边形，若沿周界绕行，每次经过一个外角，最终刚好转一整圈。因此：
Sum of exterior angles = 360°
In a regular polygon, all exterior angles are equal. So each exterior angle E = 360° ÷ n.
在正多边形中，所有外角都相等。所以每个外角 E = 360° ÷ n。
4. The 174° Question Step-by-Step | 174° 问题的分步解答
Given a regular polygon with interior angle 174°, we first find the exterior angle:
已知一个正多边形的内角为 174°，我们先求外角：
Exterior angle = 180° − 174° = 6°
Now use the rule that the sum of exterior angles is 360°. Since every exterior angle is equal:
再利用“外角和为 360°”的规律。因为每个外角相等：
n = 360° ÷ 6° = 60
So the polygon has 60 sides. This is known as a hexacontagon. The interior angle of 174° is unusually large because there are so many sides.
因此这个多边形有 60 条边，称为六十边形。内角 174° 非常大，正是因为边数很多。
5. Relationship Between Interior and Exterior Angles | 内角与外角的关系
If a regular polygon has n sides, its exterior angle E and interior angle I satisfy:
如果一个正多边形有 n 条边，它的外角 E 和内角 I 满足：
E = 360° ÷ n, I = 180° − E
Combining these gives a direct formula for the interior angle:
将两式组合，可得到内角的直接公式：
I = 180° − 360°/n
When n is large, 360°/n is small, so I approaches 180°. For a regular hexagon (n = 6), I = 120°; for a regular pentagon (n = 5), I = 108°.
当 n 很大时，360°/n 很小，所以 I 趋近于 180°。正六边形 (n = 6) 的内角为 120°，正五边形 (n = 5) 的内角为 108°。
6. Useful Table of Regular Polygons | 常见正多边形速查表
| Number of sides (n) | Exterior angle | Interior angle |
| 3 | 120° | 60° |
| 4 | 90° | 90° |
| 5 | 72° | 108° |
| 6 | 60° | 120° |
| 8 | 45° | 135° |
| 10 | 36° | 144° |
| 60 | 6° | 174° |
Notice that the 60-sided regular polygon matches the original 174° question perfectly.
注意，六十边形的数据与题目中的 174° 完全吻合。
7. Worked Examples | 典型例题详解
Example 1: A regular polygon has 12 sides. Find its interior angle.
例 1：一个正多边形有 12 条边，求它的内角。
Exterior angle = 360° ÷ 12 = 30°.
外角 = 360° ÷ 12 = 30°。
Interior angle = 180° − 30° = 150°.
内角 = 180° − 30° = 150°。
Example 2: A regular polygon has an exterior angle of 24°. How many sides does it have?
例 2：一个正多边形的外角为 24°，它有多少条边？
n = 360° ÷ 24° = 15, so it is a regular 15-gon.
n = 360° ÷ 24° = 15，所以是正十五边形。
Example 3: The sum of the interior angles of a polygon is 1620°. Find the number of sides.
例 3：一个多边形的内角和为 1620°，求边数。
Using (n − 2) × 180° = 1620°:
使用 (n − 2) × 180° = 1620°：
n − 2 = 1620° ÷ 180° = 9, so n = 11
8. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Confusing interior and exterior angles. Always identify which one is given.
错误 1：混淆内角和外角。做题时先确定已知的是哪个角。
Mistake 2: Using the formula for interior angle sum on a regular polygon without dividing by n.
错误 2：计算正多边形内角时，只算了内角和，却忘记除以边数 n。
Mistake 3: Forgetting the exterior angle sum is always 360°, even for irregular polygons.
错误 3：忽略外角和恒为 360°，即使对于不规则多边形也成立。
Useful tip: If you know the interior angle, find the exterior angle first. Then divide 360° by that number. This method is fast and reliable.
实用技巧：如果已知内角，先求外角，再用 360° 除以外角。这个方法快捷又可靠。
Always show your working, especially the step 180° − (given interior angle). Edexcel marks often reward clear method.
一定要写出过程，尤其是 180° −（已知内角）这一步。Edexcel 评分时很看重清晰的解题步骤。
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