📚 Angles in Polygons – Interior and Exterior Angles | 多边形的内角与外角
Polygons are all around us – from triangles and rectangles to hexagons found in honeycombs. Understanding how to work out the interior and exterior angles of any polygon is a core skill in the Cambridge KS3 mathematics curriculum. This topic builds on your knowledge of angles on a straight line, angles around a point, and the properties of triangles and quadrilaterals. It leads directly to the angle sum formulas and helps you solve problems involving both regular and irregular polygons, as well as real-world design and tessellation puzzles.
多边形在我们身边随处可见——从三角形、矩形到蜂巢中的六边形。掌握如何计算任意多边形的内角和外角是剑桥KS3数学课程的核心技能。这个主题基于你对平角、周角以及三角形和四边形性质的理解，直接引出了内角和公式，并帮助你解决涉及正多边形和不规则多边形的问题，以及现实中的设计和镶嵌谜题。
1. What is a Polygon? | 什么是多边形？
A polygon is a closed 2D shape made entirely of straight line segments. The segments meet only at their endpoints, called vertices. Polygons are named according to the number of sides they have: a triangle has 3 sides, a quadrilateral has 4 sides, a pentagon has 5, a hexagon has 6, and so on. In KS3, we focus on convex polygons, where all interior angles are less than 180° and all vertices point outward. Understanding the basic naming and structure is the first step towards working with angle properties.
多边形是由完全由直边组成的封闭二维图形。这些边仅在端点（称为顶点）处相交。多边形根据边数命名：三角形有3条边，四边形有4条边，五边形有5条边，六边形有6条边，依此类推。在KS3阶段，我们主要研究凸多边形，即所有内角都小于180°且所有顶点都朝外的多边形。了解基本命名和结构是学习角度性质的第一步。
2. Triangles and Interior Angles | 三角形与内角
The simplest polygon is the triangle. A key fact you must remember is that the sum of the interior angles in any triangle is always 180°. You can demonstrate this by drawing a triangle on paper, tearing off its three corners, and arranging them to form a straight line. This property works for all triangles – equilateral, isosceles, scalene, right-angled, or obtuse. It is the foundation for finding the angle sum of any polygon with more sides.
最简单的多边形是三角形。你必须记住的一个关键事实是：任何三角形的内角和总是180°。你可以通过在纸上画一个三角形，撕下它的三个角，然后把它们拼成一条直线来演示这个性质。这个性质适用于所有三角形——等边三角形、等腰三角形、不等边三角形、直角三角形或钝角三角形。它是求边数更多的任意多边形内角和的基础。
3. Quadrilaterals and Angle Sum | 四边形与角度和
Any quadrilateral can be divided into two triangles by drawing one diagonal. Since each triangle contributes 180°, the sum of the interior angles of a quadrilateral is 2 × 180° = 360°. This is true whether the shape is a square, rectangle, parallelogram, trapezium, or an irregular four-sided figure. Knowing this helps you find missing angles when three are already given: simply subtract their total from 360°.
任何一个四边形都可以通过画一条对角线分成两个三角形。由于每个三角形提供180°，所以四边形的内角和为 2 × 180° = 360°。无论形状是正方形、矩形、平行四边形、梯形还是不规则的四边形，这个结论都成立。知道这一点后，当已知三个角时，你就能求出缺失的角：只需从360°中减去已知角的总和即可。
4. The General Formula for Interior Angles | 内角和的通用公式
Extending the triangle method, an n-sided polygon can be divided into (n – 2) triangles by drawing diagonals from one vertex. Therefore, the sum of interior angles = (n – 2) × 180°. For a pentagon (n=5), the sum is 3 × 180° = 540°; for a hexagon (n=6), it is 4 × 180° = 720°. This formula works for any convex polygon, regular or irregular. It is one of the most important results in KS3 geometry and is used extensively in problem-solving.
将三角形的方法进行推广，一个 n 边形可以从一个顶点出发画对角线分成 (n – 2) 个三角形。因此，内角和 = (n – 2) × 180°。对于五边形（n=5），和为3 × 180° = 540°；对于六边形（n=6），和为4 × 180° = 720°。这个公式适用于任何凸多边形，无论正或不规则。它是KS3几何中最重要的结论之一，在解决问题时被广泛使用。
5. Exterior Angles of Polygons | 多边形的外角
An exterior angle of a polygon is formed by extending one of its sides and measuring the angle between that extension and the adjacent side. At each vertex, there are two possible exterior angles, but we normally consider the one that lies on the outside of the polygon when walking around the shape in one direction. Exterior angles are often easier to work with because they have a very simple sum rule, independently of the number of sides.
多边形的外角是通过延长它的一条边，并测量该延长线与相邻边之间的夹角而形成的。在每个顶点处有两个可能的外角，但我们通常考虑按同一方向沿多边形行走时位于外侧的那个角。外角往往更容易处理，因为它们有一个非常简单的求和规律，与边数无关。
6. Relationship Between Interior and Exterior Angles | 内角与外角的关系
At any vertex of a polygon, the interior angle and the exterior angle sit on a straight line. This means they always add up to 180°. If you know one, you can find the other instantly: Exterior angle = 180° – interior angle. This relationship is crucial when tackling problems that switch between interior and exterior angles, especially for regular polygons where we can compute each angle quickly.
在多边形的任何一个顶点处，内角和外角都位于一条直线上。这意味着它们之和总是180°。如果你知道其中一个，就能立刻求出另一个：外角 = 180° – 内角。在需要在内角和外角之间转换的问题中，这个关系至关重要，尤其是对于可以快速计算出每个角的正多边形而言。
7. Sum of Exterior Angles | 外角和
If you take one exterior angle at each vertex of any convex polygon, their total sum is always 360°. This surprising fact holds for triangles, quadrilaterals, pentagons, and any polygon no matter how many sides it has. You can imagine walking around the polygon and turning at each corner – one full turn is 360°. This property makes it very easy to find the number of sides when an exterior angle is known, or to find each exterior angle of a regular polygon.
如果你在任意凸多边形的每个顶点各取一个外角，它们的总和总是360°。这个惊人的事实适用于三角形、四边形、五边形以及任意边数的多边形。你可以想象沿着多边形行走并在每个拐角处转身——一整圈就是360°。这个性质使得当已知一个外角时很容易求出边数，或求出正多边形的每个外角。
8. Regular Polygons – Finding Each Angle | 正多边形 – 求每个角
A regular polygon has all sides equal and all angles equal. For a regular n-sided polygon, you can find the measure of each interior angle by dividing the total interior angle sum by n: each interior angle = [(n – 2) × 180°] / n. Alternatively, using the exterior angle: each exterior angle = 360° / n, and then each interior angle = 180° – exterior angle. The table below summarises these values for common regular polygons.
正多边形的所有边都相等，所有角也都相等。对于一个正 n 边形，你可以通过将内角总和除以 n 来求出每个内角的度数：每个内角 = [(n – 2) × 180°] / n。或者，利用外角：每个外角 = 360° / n，然后每个内角 = 180° – 外角。下表总结了一些常见正多边形的这些数值。
| Polygon | Sides (n) | Sum of interior angles | Each interior angle | Each exterior angle |
|---|---|---|---|---|
| Equilateral triangle | 3 | 180° | 60° | 120° |
| Square | 4 | 360° | 90° | 90° |
| Regular pentagon | 5 | 540° | 108° | 72° |
| Regular hexagon | 6 | 720° | 120° | 60° |
| Regular octagon | 8 | 1080° | 135° | 45° |
| Regular decagon | 10 | 1440° | 144° | 36° |
从上表可以看出，随着边数的增加，正多边形的每个内角变大，而每个外角变小。这两个公式让你能够轻松地在不同的表示之间切换。请记住，边数必须为整数，角度通常精确到最近的十分之一度。在剑桥KS3考试中，你需要能快速为边数不超过12的正多边形填充这样的表格。
9. Solving Problems with Polygon Angles | 用多边形角度解决问题
Many exam questions ask you to find the number of sides of a regular polygon given one of its angles. For instance, if each exterior angle is 30°, then n = 360° / 30° = 12, so it is a regular dodecagon. If the interior angle is 140°, first find the exterior angle: 180° – 140° = 40°, then n = 360° / 40° = 9, a nonagon. You might also be given the sum of interior angles and asked to find the polygon. If the sum is 1080°, solve (n-2)×180° = 1080°, so n-2 = 6, giving n = 8, an octagon.
许多考试题目会要求你在已知正多边形一个角的情况下求它的边数。例如，如果每个外角是30°，那么 n = 360° / 30° = 12，所以这是一个正十二边形。如果内角是140°，先求外角：180° – 140° = 40°，然后 n = 360° / 40° = 9，一个九边形。有时题目给出内角和让你判断多边形。如果内角和为1080°，解方程 (n-2)×180° = 1080°，得到 n-2 = 6，因此 n = 8，即八边形。
Irregular polygons can also appear. You may be given several interior angles and need to find a missing one using the sum formula. Simply calculate the total sum for that polygon using (n-2)×180°, subtract the known angles, and the remainder is the missing angle. Always check that your answer makes sense – interior angles of convex polygons must be between 0° and 180°.
不规则多边形也可能出现。题目可能给出几个内角，要求你利用求和公式找出缺失的角。只需用 (n-2)×180° 计算出该多边形的内角总和，减去已知角度，剩下的就是缺失的角。务必检查答案是否合理——凸多边形的内角必须在0°到180°之间。
10. Real-life Applications and Summary | 实际应用与总结
Angle properties of polygons appear in everyday life: tiling patterns on floors, the design of sports stadiums, and the construction of bridges all rely on exact angle calculations. Regular hexagons tessellate perfectly because their interior angle of 120° fits exactly around a point (3 × 120° = 360°). Understanding these geometric rules helps architects and engineers create stable, aesthetic structures.
多边形的角度性质出现在日常生活中：地板上的瓷砖图案、体育场的设计和桥梁的建造都依赖于精确的角度计算。正六边形可以完美镶嵌，因为其120°的内角正好可以围绕一点拼合（3 × 120° = 360°）。理解这些几何规则有助于建筑师和工程师建造稳固而美观的结构。
To summarise, in KS3 Cambridge Mathematics you must be fluent with: the (n-2)×180° rule for the sum of interior angles, the 360° sum of exterior angles, the relationship interior + exterior = 180°, and how to apply these to regular and irregular polygons. Practice by drawing diagrams and working through mixed problems until these facts become second nature.
总结来说，在剑桥KS3数学中，你必须熟练掌握：内角和的 (n-2)×180° 规则、外角和为360°、内角+外角=180°的关系，以及如何将它们应用于正多边形和不规则多边形。通过画图和做混合练习来训练，直到这些知识变成你的第二天性。
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