📚 Interior and Exterior Angles of Polygons | 多边形的内角与外角
Understanding angle properties of polygons is a core IGCSE geometry skill. It connects algebra, arithmetic and spatial reasoning. Mastering interior and exterior angle rules helps you solve both routine and multi-step problems involving regular and irregular polygons.
掌握多边形的角性质是IGCSE几何的核心技能。它将代数、算术与空间推理联系起来。熟练掌握内角与外角规则，有助于解决涉及正多边形和不规则多边形的常规题与多步题。
1. What Is a Polygon? | 什么是多边形？
A polygon is a closed two-dimensional shape made of straight sides. Triangles, quadrilaterals, pentagons and hexagons are common examples. The word polygon comes from Greek roots meaning many angles. In IGCSE, you need to know names of polygons up to at least 10 sides, such as heptagon, octagon, nonagon and decagon.
多边形是由直线边围成的封闭二维图形。三角形、四边形、五边形和六边形是常见例子。polygon一词源自希腊语，意思是许多角。在IGCSE中，你需要至少掌握到十边形的名称，如七边形、八边形、九边形和十边形。
| Number of sides | Name | 中文名称 |
|---|---|---|
| 3 | Triangle | 三角形 |
| 4 | Quadrilateral | 四边形 |
| 5 | Pentagon | 五边形 |
| 6 | Hexagon | 六边形 |
| 7 | Heptagon | 七边形 |
| 8 | Octagon | 八边形 |
| 9 | Nonagon | 九边形 |
| 10 | Decagon | 十边形 |
2. Sum of Interior Angles | 内角和
. For example, a pentagon has 5 sides, so its interior angle sum is (5 - 2) × 180° = 540°.
对于任意 n 边形，内角和为 (n - 2) × 180°。这个公式来自从一个顶点出发将多边形分割成 (n - 2) 个三角形。例如，五边形有5条边，所以内角和为 (5 - 2) × 180° = 540°。
Sum of interior angles = (n − 2) × 180°
3. Interior Angle of a Regular Polygon | 正多边形的单个内角
/ n. For a regular hexagon, one interior angle is [(6 - 2) × 180°] / 6 = 720° / 6 = 120°.
正多边形的所有边相等，所有角相等。求单个内角时，将内角和除以 n。公式为 [(n - 2) × 180°] / n。以正六边形为例，单个内角为 [(6 - 2) × 180°] / 6 = 720° / 6 = 120°。
One interior angle of a regular n-gon = [(n − 2) × 180°] ÷ n
4. Sum of Exterior Angles | 外角和
The exterior angles of any polygon, one at each vertex, always add up to 360°. This is true for both regular and irregular polygons. An exterior angle is formed by extending one side at a vertex. Because each interior angle and its adjacent exterior angle lie on a straight line, they add to 180°.
任意多边形的外角和总是360°，无论正多边形还是不规则多边形。外角是通过延长一个顶点处的一条边形成的。由于每个内角与其相邻外角在同一直线上，它们的和为180°。
Sum of exterior angles = 360°
5. Exterior Angle of a Regular Polygon | 正多边形的单个外角
exterior angle is 360° ÷ 9 = 40°. This is often the fastest route to find exterior angles.
在正多边形中，所有外角相等。因此，单个外角就是 360° ÷ n。例如，正九边形有9条边，每个外角为 360° ÷ 9 = 40°。这通常是求外角最快的途径。
One exterior angle of a regular n-gon = 360° ÷ n
6. Relationship Between Interior and Exterior Angles | 内角与外角的关系
or angle of 156°, its exterior angle is 180° - 156° = 24°; then n = 360° ÷ 24° = 15 sides.
在每个顶点处，内角与外角互补：它们的和为180°。这提供了一种快速转换方法。如果一个正多边形的内角为156°，它的外角为180° - 156° = 24°；那么 n = 360° ÷ 24° = 15 条边。
Interior angle + Exterior angle = 180°
7. Naming Angles on Parallel Lines | 平行线与截线中的角
IGCSE questions often combine polygon angles with angles on parallel lines. Key angle facts include: corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°. You may need to spot Z-shapes, F-shapes and C-shapes in diagrams.
IGCSE题目常将多边形角与平行线上的角结合。关键角事实包括：同位角相等，内错角相等，同旁内角互补为180°。你可能需要在图中识别Z形、F形和C形。
- Corresponding angles are equal | 同位角相等
- Alternate angles are equal | 内错角相等
- Co-interior angles add to 180° | 同旁内角互补为180°
8. Solving Missing Angle Problems | 求未知角问题
°, 120°, 130°, let the fifth angle be x. Then 100 + 110 + 120 + 130 + x = 540, so x = 80°.
求未知角时，首先确定多边形类型和已知角。利用内角和或直线角事实列出方程。例如一个不规则五边形已知四个角为100°、110°、120°、130°，设第五个角为 x。则 100 + 110 + 120 + 130 + x = 540，得 x = 80°。
100° + 110° + 120° + 130° + x = 540° → x = 80°
9. Common IGCSE Exam Mistakes | IGCSE常见错误
ngle of a regular polygon without dividing by n. Another is forgetting that exterior angle sum is always 360°, regardless of the number of sides. Also, students sometimes confuse interior and exterior angle formulas. Always check whether the question asks for a sum or a single angle.
常见错误是将公式 (n - 2) × 180° 直接当作正多边形单个内角，而没有除以 n。另一个错误是忘记外角和总是360°，无论边数多少。此外，学生有时会混淆内角和外角公式。一定要看清题目问的是角和还是单个角。
10. Worked Examples | 例题解析
e interior angle. Solution: n = 360° ÷ 18° = 20 sides. Interior angle = 180° - 18° = 162°.
例1：一个正多边形的外角为18°。求边数和一个内角。解：n = 360° ÷ 18° = 20 条边。内角 = 180° - 18° = 162°。
mber of sides. Solution: Exterior angle = 180° - 170° = 10°, so n = 360° ÷ 10° = 36 sides.
例2：一个正多边形的内角为170°。求边数。解：外角 = 180° - 170° = 10°，所以 n = 360° ÷ 10° = 36 条边。
11. Exam Tips Summary | 考试技巧总结
Memorise the two key formulas and the 360° exterior sum. Show your working clearly, especially when writing equations. In regular polygon problems, use exterior angles to find the number of sides quickly. Always label unknowns and check answers by substitution.
记住两个关键公式和外角和360°。写出清晰的解题过程，尤其是列方程时。在正多边形问题中，利用外角快速求边数。始终标记未知量，并通过代入检验答案。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图