📚 108 Degrees: Understanding Polygon Angles | 108度：理解多边形角度
The number 108 is not just a number — it is the exact size, in degrees, of each interior angle of a regular pentagon. This special angle appears in geometry, art, nature, and even the golden ratio. For IGCSE Edexcel Mathematics students, understanding 108° means grasping the core rules of polygon angles, exterior angle sums, and tessellation. In this article, we unpack everything you need to know about 108° and polygon geometry, with clear explanations, worked examples, and exam tips.
数字108不仅仅是一个数字——它是正五边形每个内角的确切度数。这个特殊的角度出现在几何、艺术、自然甚至黄金比例中。对于IGCSE Edexcel数学学生来说，理解108°意味着掌握多边形角度、外角和以及镶嵌的核心规则。在本文中，我们将解读你关于108°和多边形几何需要了解的全部内容，包括清晰的解释、例题和考试技巧。
1. The Angle 108° in a Pentagon | 五边形中的108°角
The first time most students encounter 108° is when calculating the interior angle of a regular pentagon. A regular pentagon has five equal sides and five equal interior angles, each measuring exactly 108°. This value is not random — it follows directly from the interior angle sum formula and makes the pentagon remarkably different from triangles (60°), squares (90°), or hexagons (120°).
大多数学生第一次遇到108°是在计算正五边形的内角时。正五边形有五条相等的边和五个相等的内角，每个内角精确地为108°。这个值不是随机的——它直接从内角和公式得出，使得五边形显著不同于三角形（60°）、正方形（90°）或六边形（120°）。
Once you know that a regular pentagon has 108° angles, you can unlock many properties: its exterior angle is 72°, its diagonals form a pentagram with 36°-72°-72° isosceles triangles, and it is intimately linked to the golden ratio φ.
一旦知道了正五边形的角是108°，许多性质就会迎刃而解：其外角为72°，对角线构成包含36°-72°-72°等腰三角形的五角星形，并且它与黄金比例φ密切相关。
2. Basic Polygon Terminology | 基本多边形术语
Before diving deeper, let's recall the essential vocabulary for any polygon (a closed 2D shape with three or more straight sides).
在深入之前，让我们回顾一下任何多边形（由三条或更多直边围成的闭合二维图形）的基本词汇。
Key terms:
- Side: one of the straight line segments that form the boundary.
- Vertex (plural: vertices): a point where two sides meet.
- Interior angle: the angle inside the polygon at a vertex.
- Exterior angle: the angle formed outside the polygon when one side is extended, supplementary to the interior angle.
- Regular polygon: a polygon with all sides equal and all interior angles equal.
- Irregular polygon: sides and/or angles are not all equal.
关键术语：
- 边：构成边界的直线段之一。
- 顶点：两条边相交的点。
- 内角：多边形内部在顶点的角。
- 外角：当一条边延长时，在多边形外形成的角，与内角互补。
- 正多边形：所有边都相等，所有内角都相等的多边形。
- 不规则多边形：边和/或角不完全相等的多边形。
3. Sum of Interior Angles Formula | 内角和公式
. This works because you can split a polygon into (n − 2) triangles by drawing diagonals from one vertex, and each triangle contributes 180°.
任意n边形的内角和总是 (n − 2) × 180°。之所以成立，是因为通过从一个顶点画对角线可以将多边形分割成 (n − 2) 个三角形，每个三角形贡献180°。
Sum of interior angles = (n − 2) × 180°
total is shared among the five interior angles, setting the stage for the regular pentagon's 108°.
对于五边形（n = 5），内角和为 (5 − 2) × 180° = 3 × 180° = 540°。这个总度数由五个内角分享，为正五边形的108°埋下伏笔。
4. Calculating the Interior Angle of a Regular Polygon | 计算正多边形的内角
gon, all interior angles are equal. Therefore, each interior angle = ((n − 2) × 180°) / n.
在正多边形中，所有内角都相等。因此，每个内角 = ((n − 2) × 180°) / n。
Interior angle (regular) = ((n − 2) × 180°) / n
For n = 5: ((5 − 2) × 180°) / 5 = (3 × 180°) / 5 = 540° / 5 = 108°.
对于n = 5：((5 − 2) × 180°) / 5 = (3 × 180°) / 5 = 540° / 5 = 108°。
The table below shows interior angles for common regular polygons. Notice how 108° sits between the square's 90° and the hexagon's 120°.
下表展示了常见正多边形的内角。注意108°如何在正方形的90°和六边形的120°之间。
| n (sides) | Name | Sum of interior angles | Each interior angle (regular) |
|---|---|---|---|
| 3 | Triangle | 180° | 60° |
| 4 | Quadrilateral | 360° | 90° |
| 5 | Pentagon | 540° | 108° |
| 6 | Hexagon | 720° | 120° |
| 8 | Octagon | 1080° | 135° |
| 10 | Decagon | 1440° | 144° |
5. Exterior Angles and the 108° Connection | 外角与108°的联系
ementary). For any convex polygon, the sum of the exterior angles (one at each vertex) is always 360°, no matter how many sides.
获得108°的一个更简单的方法是利用外角。多边形的外角是通过延长一条边形成的；在每个顶点，内角 + 外角 = 180°（它们互补）。对于任何凸多边形，外角之和（每个顶点一个）总是360°，与边数无关。
Sum of exterior angles = 360°
e = 180° − 72° = 108°. This method is often faster and avoids dealing with (n − 2) × 180°.
在正多边形中，每个外角 = 360° / n。对于正五边形，外角 = 360° / 5 = 72°。然后内角 = 180° − 72° = 108°。这种方法通常更快，并且避免了 (n − 2) × 180° 的计算。
terior angle of 108°, then n = 360° / (180° − 108°) = 360° / 72° = 5. So it is a pentagon.
你可以利用外角技巧，在给定内角时求边数。例如，如果一个正多边形的内角是108°，那么 n = 360° / (180° − 108°) = 360° / 72° = 5。所以这是一个五边形。
6. Why 108° Matters: Geometry and Nature | 108°为何重要：几何与自然
The 108° angle appears far beyond textbooks. Many flowers, such as morning glories and certain lilies, have a pentagonal petal arrangement. The starfish (sea star) typically has five arms radiating at 72° angles from the center, creating inner angles of 108° between its arms. Even the pattern of some viruses and marine organisms is based on pentagonal symmetry.
108°角远不止出现在课本中。许多花朵，如牵牛花和某些百合，都呈五边形的花瓣排列。海星通常有五条臂，从中心以72°角辐射，臂之间的内角为108°。甚至某些病毒和海洋生物的结构图案也是基于五边形对称。
Architecturally, the Pentagonal fortress designs and the famous Pentagon building in the USA echo this geometry. Because 108° is linked to φ (the golden ratio), it appears in aesthetically pleasing designs and Renaissance art.
在建筑方面，五边形堡垒设计以及美国著名的五角大楼都呼应了这一几何。因为108°与黄金比例 φ 相连，它出现在美学上令人愉悦的设计和文艺复兴时期的艺术中。
7. The Golden Ratio in the Pentagon | 五边形中的黄金比例
A regular pentagon hides the golden ratio φ = (1 + √5 nals of a regular pentagon, you form a pentagram (five-pointed star). The diagonals intersect each other in golden ratio proportions: the ratio of a diagonal to a side of the pentagon is φ.
正五边形隐藏着黄金比例 φ = (1 + √5) / 2 ≈ 1.618。如果你画出正五边形的所有五条对角线，就会形成一个五角星形（五角星）。对角线彼此交叉形成黄金比例：对角线长度与五边形边长之比等于 φ。
−cos 72° and express it with φ. This connection is beyond IGCSE but shows the depth of 108° in higher mathematics.
内角108°产生了与 φ 优美相关的三角值。例如，cos 36° = φ / 2，sin 18° = (φ − 1) / 2。因为108° = 180° − 72°，你可以将 cos 108° 与 −cos 72° 联系起来，并用 φ 表示。这种联系虽超出IGCSE范围，但展示了108°在高等数学中的深度。
8. Worked Example: Finding Angles in Polygons | 例题：求多边形角度
Example 1: An irregular pentagon has four interior angles of 110°, 95°, 130° and 85°. Find the fifth angle.
例题1：一个不规则五边形有四个内角分别是110°、95°、130°和85°。求第五个角。
Sum of interior angles of a pentagon = (5 − 2) × 180° = 540°.
Sum of known angles = 110° + 95° + 130° + 85° = 420°.
Fifth angle = 540° − 420° = 120°.
五边形内角和 = (5 − 2) × 180° = 540°。
已知角之和 = 110° + 95° + 130° + 85° = 420°。
第五个角 = 540° − 420° = 120°。
Example 2: A regular polygon has an interior angle of 108°. How many sides does it have?
例题2：一个正多边形的内角是108°。它有多少条边？
Method 1: Exterior angle = 180° − 108° = 72°. n = 360° / 72° = 5 sides.
iply both sides by n: (n − 2) × 180° = 108°n → 180°n − 360° = 108°n → 72°n = 360° → n = 5.
方法1：外角 = 180° − 108° = 72°。n = 360° / 72° = 5条边。
方法2：((n − 2) × 180°) / n = 108° → 两边同乘 n：(n − 2) × 180° = 108°n → 180°n − 360° = 108°n → 72°n = 360° → n = 5。
9. Tessellations: Can a Pentagon Tile the Plane? | 镶嵌：五边形能铺满平面吗？
A tessellation (tiling) occurs when identical shapes fit together without gaps or overlaps, using only their copies. For regular polygons to tessellate, the interior angle must be a divisor of 360°, because the angles around a point where tiles meet must sum to 360°.
当相同的形状无缝隙、不重叠地拼接在一起时，就形成了镶嵌（平铺）。要使正多边形能够镶嵌，内角必须是360°的约数，因为在拼块相交点的周围角度之和必须为360°。
ur pentagons give 432°, which overlaps. Therefore, a regular pentagon cannot tessellate the plane on its own. In contrast, equilateral triangles (60°), squares (90°), and regular hexagons (120°) all tessellate perfectly.
正五边形的内角 = 108°。检查：360° / 108° = 3.333…，不是整数。如果尝试在一个点周围放置三个五边形，总角度 = 3 × 108° = 324°，留下36°的缝隙。四个五边形则得到432°，会重叠。因此，正五边形单独不能铺满平面。相比之下，等边三角形（60°）、正方形（90°）和正六边形（120°）都能完美镶嵌。
Irregular pentagons can tile the plane (there are 15 known types of pentagonal tilings), but the regular pentagon's 108° angle makes it impossible.
不规则五边形可以铺满平面（已知有15种五边形平铺类型），但正五边形的108°角使其不可能。
10. Exam-Style Questions | 考试风格题目
Q1 (Edexcel IGCSE style): The interior angle of a regular polygon is 108°. Work out the number of sides of this polygon.
问题1（Edexcel IGCSE风格）：一个正多边形的内角是108°。计算这个多边形的边数。
Answer: Exterior angle = 180° − 108° = 72°. Number of sides = 360° ÷ 72° = 5.
答案：外角 = 180° − 108° = 72°。边数 = 360° ÷ 72° = 5。
Q2: Show that the sum of the interior angles of a pentagon is 540°, and hence explain why each interior angle of a regular pentagon is 108°.
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