Angles in Polygons | 多边形中的角

📚 Angles in Polygons | 多边形中的角

Polygons are everywhere in geometry, and understanding how to calculate their angles is a core skill for IGCSE Mathematics. This article breaks down the rules of interior and exterior angles step by step, with worked examples and common pitfalls.

多边形几何无处不在，掌握多边形角度计算是 IGCSE 数学的核心技能。本文将逐步讲解内角和外角的规律，配以典型例题和常见易错点，帮助你轻松应对考试。

1. What Is a Polygon? | 什么是多边形？

A polygon is a closed 2D shape made of straight line segments. Examples include triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), and octagons (8 sides).

多边形是由若干条直线段围成的封闭二维图形。例如三角形（3条边）、四边形（4条边）、五边形（5条边）、六边形（6条边）和八边形（8条边）。

A polygon with n sides has n vertices.
有 n 条边的多边形有 n 个顶点。
The sum of angles depends only on the number of sides, not on the shape.
多边形的角度总和只与边数有关，与具体形状无关。

2. Interior and Exterior Angles | 内角与外角

An interior angle is the angle formed inside the polygon between two adjacent sides. An exterior angle is formed by extending one side outward; it lies outside the polygon.

内角是两条相邻边在图形内部形成的角。外角是由一条边延长后与邻边在图形外部形成的角。

At each vertex, the interior angle and the exterior angle add up to 180°.
在每个顶点处，内角与外角之和为 180°。
If the interior angle is I and the exterior angle is E, then I + E = 180°.
若内角为 I、外角为 E，则 I + E = 180°。

3. The Interior Angle Sum Formula | 内角和公式

For any polygon with n sides, the sum of its interior angles is given by the formula:

Sum of interior angles = (n − 2) × 180°

This works because any polygon can be divided into (n − 2) triangles from one vertex, and each triangle contributes 180°.

这是因为从多边形的一个顶点出发，可以将它分成 (n − 2) 个三角形，而每个三角形的内角和为 180°。

Number of sides (n)	Sum of interior angles
4	(4−2)×180° = 360°
5	(5−2)×180° = 540°
8	(8−2)×180° = 1080°

An octagon (8 sides) has an interior angle sum of 1080°, which is exactly the number in this article’s title.

八边形（8条边）的内角和为 1080°，正是本文标题中的数字。

4. Regular Polygons | 正多边形

A regular polygon has all sides equal in length and all interior angles equal in size. For a regular polygon, we can find each interior angle by dividing the total sum by the number of sides.

正多边形的所有边长相等，所有内角大小也相等。对于正多边形，可以将内角和除以边数，得到每个内角的大小。

Each interior angle of a regular n-gon = ((n − 2) × 180°) ÷ n

Polygon	n	Each interior angle
Regular triangle	3	60°
Regular square	4	90°
Regular pentagon	5	108°
Regular hexagon	6	120°
Regular octagon	8	135°

5. The Exterior Angle Sum Theorem | 外角和定理

The sum of all exterior angles of any convex polygon is always 360°, regardless of the number of sides. In a regular polygon, each exterior angle is equal to 360° ÷ n.

任何凸多边形的外角和恒为 360°，与边数无关。在正多边形中，每个外角等于 360° ÷ n。

Sum of exterior angles = 360°

Each exterior angle (regular n-gon) = 360° ÷ n

This theorem is extremely useful for finding n when only an exterior angle is given.
这个定理在已知外角求边数时非常实用。
Remember: exterior and interior angles at the same vertex are supplementary.
记住：同一顶点处的外角与内角互补（和为 180°）。

6. Finding the Number of Sides | 求多边形的边数

If you know the interior angle sum or the measure of each exterior angle, you can determine the number of sides of the polygon.

如果你知道内角和或每个外角的大小，就可以确定多边形的边数。

Example 1: A polygon has an interior angle sum of 1260°. Find n.

例 1：一个多边形的内角和为 1260°，求 n。

(n − 2) × 180 = 1260

n − 2 = 1260 ÷ 180 = 7

n = 9

So the polygon has 9 sides.

因此这个多边形有 9 条边。

Example 2: A regular polygon has an exterior angle of 40°. How many sides does it have?

例 2：一个正多边形的外角为 40°，它有多少条边？

n = 360 ÷ 40 = 9

Again, the polygon has 9 sides.

同样，该多边形有 9 条边。

7. Solving for Unknown Angles | 求解未知角

In IGCSE questions, you may be given a diagram with some known angles and one unknown angle. Use the interior angle sum to set up an equation and solve.

在 IGCSE 题目中，你可能会看到带有若干已知角和未知角的图形。利用内角和建立方程求解即可。

Example 3: A quadrilateral has three angles: 80°, 95°, and 120°. Find the fourth angle.

例 3：一个四边形有三个角：80°、95° 和 120°，求第四个角。

The sum of interior angles in a quadrilateral is 360°.

四边形的内角和为 360°。

360 − (80 + 95 + 120) = 360 − 295 = 65°

Therefore the fourth angle is 65°.

因此第四个角为 65°。

8. Working with Regular Hexagons and Octagons | 正六边形与正八边形

Regular hexagons (each 120°) and regular octagons (each 135°) appear frequently in tessellation and angle-chasing problems.

正六边形（每个角 120°）和正八边形（每个角 135°）经常出现在镶嵌和角度推理题中。

Example 4: Three regular hexagons meet at a point. What is the total angle around the point?

例 4：三个正六边形在一个点相交，该点周围的总角度是多少？

Each interior angle is 120°, so three of them give 3 × 120° = 360°.

每个内角为 120°，三个内角为 3 × 120° = 360°。

This is why regular hexagons tile a plane perfectly.

这就是正六边形能够完美平铺平面的原因。

9. Composite Angle Problems | 复合角度问题

Some questions combine polygons with triangles or parallel lines. Always look for straight lines (180°), triangles (180°), and full turns (360°) in the diagram.

有些题目会将多边形与三角形或平行线结合。解题时要留意图中的平角（180°）、三角形（180°）和圆周角（360°）。

Example 5: A regular pentagon is placed next to a regular hexagon at a common vertex. Find the angle between one side of the pentagon and one side of the hexagon that meet at that vertex.

例 5：一个正五边形与一个正六边形共用顶点拼接，求该顶点处五边形一条边与六边形一条边之间的夹角。

Interior angle of pentagon = 108°; interior angle of hexagon = 120°. The full turn around the vertex is 360°, so the unknown angle is:

正五边形内角为 108°；正六边形内角为 120°。顶点一周为 360°，因此未知夹角为：

360 − 108 − 120 = 132°

Always subtract the two interior angles from 360° because the three angles meet at a point.

因为三个角交于一点，所以要用 360° 减去两个内角。

10. Common Mistakes and Tips | 常见错误与技巧

Students often confuse interior and exterior angles or forget the (n − 2) factor. The following table summarises the key facts.

学生经常混淆内角与外角，或者忘记 (n − 2) 这个因子。下表总结了关键要点。

Case	Formula
Sum of interior angles of n-gon	(n − 2) × 180°
Each interior angle of regular n-gon	((n − 2) × 180°) ÷ n
Sum of exterior angles (any polygon)	360°
Each exterior angle of regular n-gon	360° ÷ n

Always check whether the polygon is regular or irregular before applying the ‘each angle’ rule.
在应用“每个角”的规则前，先确认多边形是正多边形还是不规则多边形。
If you know each interior angle, first find the exterior angle: 180° − interior angle.
若已知每个内角，先求外角：180° − 内角。
Use 360° ÷ exterior angle to find n quickly for regular polygons.
对于正多边形，用 360° ÷ 外角可以快速求出 n。
Do not use 360° for interior angle sum — that only works for quadrilaterals.
不要把 360° 当作内角和公式 — 那仅适用于四边形。

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Angles in Polygons | 多边形中的角