📚 Interior and Exterior Angles of Polygons | 多边形的内角与外角
Understanding the angles inside and outside polygons is a core IGCSE Mathematics skill. In this revision guide, you will learn the angle-sum formulas, practise with regular and irregular polygons, and see why an octagon has a total interior angle sum of 1080°.
理解多边形内角与外角是 IGCSE 数学的核心技能。在本复习指南中，你将学习角度和公式，练习正多边形与不规则多边形，并理解为什么八边形的内角和为 1080°。
1. Polygon Basics | 多边形基础
A polygon is a closed two-dimensional shape made entirely from straight sides. The word 'regular' means all side lengths are equal and all interior angles are equal; otherwise the polygon is irregular.
多边形是一个完全由直线边组成的封闭二维图形。"正" 多边形意味着所有边长相等且所有内角相等；否则就是不规则多边形。
In IGCSE problems, you will usually work with convex polygons, where every interior angle is less than 180°. Common examples include triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons and decagons.
在 IGCSE 题目中，你通常会遇到凸多边形，即每个内角都小于 180°。常见例子包括三角形、四边形、五边形、六边形、七边形、八边形、九边形和十边形。
o on. In an irregular polygon, sides and angles may all be different, but the total angle sums still follow fixed rules.
边数通常写作 n。三角形 n = 3，四边形 n = 4，五边形 n = 5，依此类推。在不规则多边形中，边长和角度可能各不相同，但总角度和仍然遵循固定规律。
- Triangle, n = 3 | 三角形，n = 3
- Quadrilateral, n = 4 | 四边形，n = 4
- Pentagon, n = 5 | 五边形，n = 5
- Hexagon, n = 6 | 六边形，n = 6
- Heptagon, n = 7 | 七边形，n = 7
- Octagon, n = 8 | 八边形，n = 8
- Nonagon, n = 9 | 九边形，n = 9
- Decagon, n = 10 | 十边形，n = 10
2. Why the Interior Angle Sum Formula Works | 内角和公式的原理
For any n-sided polygon, the sum of all interior angles is found by subtracting 2 from the number of sides and multiplying by 180°.
对于任意 n 边形，所有内角之和等于边数减 2 后乘以 180°。
Sum of interior angles = (n − 2) × 180°
This formula comes from dividing the polygon into (n − 2) triangles, each of which has an interior angle sum of 180°.
这个公式来自把多边形分割成 (n − 2) 个三角形；每个三角形内角和为 180°。
s divides into exactly n − 2 triangles, so the total interior angle sum is (n − 2) × 180°.
选择任意一个顶点，向所有不相邻的顶点画对角线。这些对角线会把多边形分割成若干三角形。n 边形总能分成正好 n − 2 个三角形，所以内角和为 (n − 2) × 180°。
a quadrilateral has n = 4, so its interior angle sum is (4 − 2) × 180° = 2 × 180° = 360°.
例如，四边形 n = 4，所以它的内角和为 (4 − 2) × 180° = 2 × 180° = 360°。
540°. A hexagon has n = 6, so its interior angle sum is (6 − 2) × 180° = 4 × 180° = 720°.
五边形 n = 5，所以内角和为 (5 − 2) × 180° = 3 × 180° = 540°。六边形 n = 6，所以内角和为 (6 − 2) × 180° = 4 × 180° = 720°。
3. Using the Interior Angle Sum Formula | 内角和公式的应用
to (n − 2) × 180°. Always write down the substitution so the examiner can see your method.
使用公式时，先确定边数 n，再代入 (n − 2) × 180°。一定要写出代入过程，让考官看到你的方法。
The table below lists common interior angle sums that appear frequently in IGCSE questions.
下表列出了 IGCSE 题目中常见的多边形内角和。
| Polygon | 多边形 | n | Interior angle sum | 内角和 |
|---|---|---|
| Triangle | 三角形 | 3 | 180° |
| Quadrilateral | 四边形 | 4 | 360° |
| Pentagon | 五边形 | 5 | 540° |
| Hexagon | 六边形 | 6 | 720° |
| Heptagon | 七边形 | 7 | 900° |
| Octagon | 八边形 | 8 | 1080° |
| Nonagon | 九边形 | 9 | 1260° |
| Decagon | 十边形 | 10 | 1440° |
angles of a pentagon are 100°, 110°, 120° and 130°, the fifth angle is 540° − 460° = 80°.
如果多边形不规则，仍然可以使用同一个总和。例如，五边形的四个角为 100°、110°、120°、130°，则第五个角为 540° − 460° = 80°。
4. Worked Example: The Octagon and 1080° | 例题：八边形与 1080°
An octagon has n = 8 sides. Substitute n = 8 into the interior angle sum formula:
八边形有 n = 8 条边。将 n = 8 代入内角和公式：
Sum = (8 − 2) × 180° = 6 × 180° = 1080°
Therefore, any octagon has a total interior angle sum of 1080°. This is why a question about 1080° usually asks you to identify or work with an octagon.
因此，任何八边形的内角和都是 1080°。这就是为什么涉及 1080° 的题目通常会让你识别或计算八边形。
f the interior angle sum is 1080°, solve (n − 2) × 180 = 1080, giving n − 2 = 6, so n = 8.
你也可以反过来使用公式：如果内角和为 1080°，解 (n − 2) × 180 = 1080，得到 n − 2 = 6，所以 n = 8。
0° ÷ 8 = 135°, but an irregular octagon can have different angles that still add to 1080°.
记住，1080° 这个总和与八边形是否为正多边形无关。正八边形有八个相等内角，每个为 1080° ÷ 8 = 135°；不规则八边形的各个角可以不同，但总和仍为 1080°。
5. Working Backwards from the Angle Sum | 由内角和反求边数
Many exam questions give the total interior angle sum and ask for the number of sides. Rearrange the formula to make n the subject.
许多考试题给出内角和，要求求边数。此时需要把公式变形，使 n 成为主项。
n = (Sum ÷ 180°) + 2
ior angle sum is 900°, then n = (900 ÷ 180) + 2 = 5 + 2 = 7, so the polygon is a heptagon.
例如，若内角和为 900°，则 n = (900 ÷ 180) + 2 = 5 + 2 = 7，所以这个多边形是七边形。
r angle sum is 1440°, then n = (1440 ÷ 180) + 2 = 8 + 2 = 10, so the polygon is a decagon.
若内角和为 1440°，则 n = (1440 ÷ 180) + 2 = 8 + 2 = 10，所以这个多边形是十边形。
This reverse method is also useful when a question says 'the interior angles of a polygon add up to 1800°'. Divide 1800 by 180 to get 10, then add 2 to get n = 12. The polygon has 12 sides.
当题目说 "多边形内角和为 1800°" 时，这个反求方法也很有用。用 1800 除以 180 得 10，再加 2 得 n = 12。这个多边形有 12 条边。
Always check that n is a whole number greater than 2. If you get a decimal or a value less than 3, you have probably made an arithmetic mistake.
一定要检查 n 是否为大于 2 的整数。如果得到小数或小于 3 的值，通常是计算出错了。
6. Exterior Angles of a Polygon | 多边形的外角
An exterior angle of a polygon is formed by extending one side and measuring the angle between the extended side and the next side. At any vertex of a convex polygon, the interior angle and the exterior angle lie on a straight line, so they add to 180°.
多边形的外角是通过延长一条边，并测量延长线与下一条边之间的夹角形成的。在凸多边形的任意顶点处，内角与外角在同一直线上，所以它们相加等于 180°。
Interior angle + Exterior angle = 180°
This relationship is true at every vertex. If you know one exterior angle, subtract it from 180° to find the interior angle at the same vertex. If you know the interior angle, subtract it from 180° to find the exterior angle.
这个关系在每个顶点都成立。如果知道一个外角，就用 180° 减去它，得到同一顶点的内角。如果知道内角，就用 180° 减去它，得到外角。
For a regular polygon, all exterior angles are equal. The exterior angle is often easier to work with than the interior angle because the total exterior angle sum is always 360°, no matter how many sides the polygon has.
对于正多边形，所有外角都相等。外角通常比内角更容易计算，因为无论多边形有多少条边，外角和始终是 360°。
In irregular polygons, exterior angles may be different sizes, but there is still one exterior angle at each vertex, and their total sum is still 360°.
在不规则多边形中，各个外角大小可能不同，但每个顶点仍然有一个外角，它们的总和仍然是 360°。
7. Exterior Angle Sum = 360° | 外角和等于 360°
The exterior angles of any convex polygon add up to 360°. This is one of the most useful facts in polygon geometry.
任意凸多边形的外角和都等于 360°。这是多边形几何中最有用的事实之一。
Sum of exterior angles = 360°
One way to see this is to imagine walking around the polygon. At each vertex you turn through the exterior angle. After walking all the way around and returning to your starting direction, you have made one full turn, which is 360°.
一种理解方法是想象绕多边形走一圈。在每个顶点你转过一个外角。走完一整圈并回到起始方向时，你总共转了一整圈，也就是 360°。
rs is n × 180°. But the total interior sum is (n − 2) × 180°. Therefore the total exterior sum is n × 180° − (n − 2) × 180° = 2 × 180° = 360°.
也可以用内角和来证明。在 n 个顶点处，内角 + 外角 = 180°。所以所有内角与所有外角的总和是 n × 180°。但内角和为 (n − 2) × 180°。因此外角和为 n × 180° − (n − 2) × 180° = 2 × 180° = 360°。
This result works for triangles, quadrilaterals, pentagons, octagons and every other convex polygon. It does not depend on whether the polygon is regular.
这个结果适用于三角形、四边形、五边形、八边形以及所有其他凸多边形。它与多边形是否为正多边形无关。
8. Regular Polygons: One Interior and One Exterior Angle | 正多边形：单个内角与外角
In a regular polygon, all interior angles are equal and all exterior angles are equal. Since the exterior angles add to 360°, each exterior angle is 360° ÷ n.
在正多边形中，所有内角相等，所有外角也相等。由于外角和为 360°，每个外角为 360° ÷ n。
Each exterior angle = 360° ÷ n
r angle of a regular polygon is 180° − 360° ÷ n. This is the same as ((n − 2) × 180°) ÷ n.
由于每个顶点处内角 + 外角 = 180°，正多边形的每个内角为 180° − 360° ÷ n。这也等于 ((n − 2) × 180°) ÷ n。
Each interior angle = 180° − (360° ÷ n)
= 5. Each exterior angle is 360° ÷ 5 = 72°, and each interior angle is 180° − 72° = 108°.
正五边形 n = 5。每个外角为 360° ÷ 5 = 72°，每个内角为 180° − 72° = 108°。
= 6. Each exterior angle is 360° ÷ 6 = 60°, and each interior angle is 180° − 60° = 120°.
正六边形 n = 6。每个外角为 360° ÷ 6 = 60°，每个内角为 180° − 60° = 120°。
| Regular polygon | 正多边形 | n | Exterior angle | 外角 | Interior angle | 内角 |
|---|---|---|---|
| Equilateral triangle | 等边三角形 | 3 | 120° | 60° |
| Square | 正方形 | 4 | 90° | 90° |
| Regular pentagon | 正五边形 | 5 | 72° | 108° |
| Regular hexagon | 正六边形 | 6 | 60° | 120° |
| Regular octagon | 正八边形 | 8 | 45° | 135° |
| Regular decagon | 正十边形 | 10 | 36° | 144° |
9. Worked Examples with Regular Polygons | 正多边形例题
360° ÷ 12 = 30°. Then subtract from 180°: 180° − 30° = 150°. Each interior angle is 150°.
例 1：求有 12 条边的正十二边形的每个内角。先求外角：360° ÷ 12 = 30°。再用 180° 减去它：180° − 30° = 150°。每个内角为 150°。
mber of sides. Use n = 360° ÷ exterior angle = 360 ÷ 24 = 15, so the polygon has 15 sides.
例 2：一个正多边形的外角为 24°。求边数。使用 n = 360° ÷ 外角 = 360 ÷ 24 = 15，所以该多边形有 15 条边。
exterior angle is 180° − 160° = 20°. Then n = 360 ÷ 20 = 18, so the polygon has 18 sides.
例 3：一个正多边形的内角为 160°。求边数。外角为 180° − 160° = 20°。然后 n = 360 ÷ 20 = 18，所以该多边形有 18 条边。
Example 4: The interior angle of a regular polygon is 8 times its exterior angle. Let the exterior angle be x. Then the interior angle is 8x, and x +
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