# 多边形

> 爱德思国际普通中等教育证书数学A · 4MA1
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s4-polygons/

本指南覆盖爱德思IGCSE数学A（4MA1）考试涉及的全部多边形相关内容，包括图形命名、四边形性质、内角/外角计算以及多边形的全等判定规则。

**先修:** [基础角度规则（平角、周角相关性质）](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s4-basic-angle-rules/); [三角形的相关性质](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s4-triangles/)

## 学习目标

- 命名并识别常见多边形与特殊四边形
- 使用公式$(n-2) \times 180^\circ$计算任意多边形的内角和
- 求解正多边形的内角与外角
- 运用特殊四边形的性质求解角度与边长相关问题
- 定义多边形的全等关系并识别全等图形

## 识别多边形与特殊四边形

**多边形** — 由3条及以上互不相交的直边构成、没有曲边的封闭二维图形。

*例:* Triangle (3 sides), pentagon (5 sides), octagon (8 sides)

- Named polygons you must recognise: pentagon (5 sides), hexagon (6 sides), octagon (8 sides)
- All 4-sided polygons are called quadrilaterals, with interior angles summing to 360°

**四边形** — 恰好拥有4条边的多边形，其内角和为360°

| Quadrilateral | Key Defining Properties |
| --- | --- |
| Parallelogram | Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other |
| Rectangle | Parallelogram with all angles 90°; diagonals equal in length |
| Square | Rectangle with all sides equal; diagonals perpendicular, bisect vertex angles |
| Rhombus | Parallelogram with all sides equal; diagonals perpendicular, bisect vertex angles |
| Trapezium | Exactly one pair of parallel opposite sides |
| Kite | Two pairs of adjacent equal sides; one pair of opposite equal angles |

**例题:** 请说出拥有两组邻边相等、且有一组对角相等的四边形的名称。

1. 1. 将给出的性质与各类四边形的定义进行比对
2. 2. 平行四边形是*对边*相等，不符合该条件
3. 3. 筝形的定义就是拥有两组邻边相等，且有一组对角相等
4. 最终答案：筝形

> **考试提示:** 考试中如果要求你命名一个四边形，一定要列出2-3条关键性质来佐证你的答案，才能拿到全部分数。

*计算器:* allowed

## 多边形的内角和

任意$n$边形的内角和都可以通过下方公式计算，该公式的推导逻辑是将多边形拆分为$n-2$个三角形，每个三角形的内角和均为180°。

$$Sum\ of\ interior\ angles = (n - 2) \times 180^\circ = (2n - 4) \times 90^\circ$$

**例题:** 计算七边形的内角和。

1. 1. 确定边数：$n=7$
2. 2. 代入公式计算：
3. $$Sum = (7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ$$
4. 最终答案：900°

**例题:** 一个四边形的三个内角分别为72°、105°和121°，求第四个内角的大小。

1. 1. 四边形的内角和为360°
2. $$2. Sum of given angles = 72 + 105 + 121 = 298^\circ$$
3. $$3. Missing angle = 360 - 298 = 62^\circ$$
4. 最终答案：62°

> **考试提示:** 做四边形角度相关题目时，你既可以代入通用公式取$n=4$计算，也可以直接使用固定的360°内角和结论来节省时间。

*计算器:* allowed

## 正多边形：内角与外角

**正多边形** — 所有边长相等、所有内角大小也相等的多边形。

*例:* Square, regular hexagon, regular octagon

正多边形的两条核心规则：<br>1. *任意*多边形（无论正或不规则）的外角和恒为360°，因此正$n$边形的每个外角 = $\frac{360^\circ}{n}$ <br>2. 每个内角的大小等于180°减去同一顶点处的外角，也可以用公式$\frac{(n-2) \times 180^\circ}{n}$计算。

**例题:** 计算正八边形每个内角的大小。

1. 1. 八边形有8条边，因此 $n=8$
2. 2. 使用更简便的外角法：
3. $$Exterior\ angle = \frac{360^\circ}{8} = 45^\circ$$
4. $$Interior\ angle = 180^\circ - 45^\circ = 135^\circ$$
5. 最终答案：135°

**例题:** 一个正多边形的内角为150°，求它有多少条边？

1. 1. 首先计算同一顶点处的外角：
2. $$Exterior\ angle = 180^\circ - 150^\circ = 30^\circ$$
3. 2. 对外角公式进行变形，求解n：
4. $$n = \frac{360^\circ}{exterior\ angle} = \frac{360}{30} = 12$$
5. 最终答案：12条边（十二边形）

> **考试提示:** 如果题目给出正多边形的一个内角、要求计算边数，建议优先使用外角相关公式求解，步骤更少，能降低计算出错的概率。

*计算器:* allowed

## 多边形的全等

**全等多边形** — 形状和大小完全相同，所有对应边和对应角都相等的多边形。图形的旋转或反射不会改变其全等性质。

*例:* Two squares with side length 5cm are congruent; a square of side 3cm and a square of side 6cm are not congruent.

**例题:** 判断边长分别为7cm和8cm的两个等边三角形是否全等，并说明理由。

1. 1. 检查二者的形状和大小是否一致：
2. 2. 二者都是等边三角形，因此形状相同，但边长不等，因此大小不同。
3. 最终答案：不全等，二者的对应边长度不相等。

> **考试提示:** 考试中解释全等概念时，必须同时提及形状相同和大小相等（或对应边/对应角全部相等）才能获得满分。

*计算器:* allowed

## 常见错误

- **错误做法:** Assuming trapeziums have two pairs of parallel sides
  - 原因: Only parallelograms have two pairs of parallel sides; trapeziums are defined as having exactly one pair of parallel sides
  - 正确做法: Quote exactly one pair of parallel sides when defining or identifying a trapezium
- **错误做法:** Using the wrong number of sides $n$ for named polygons
  - 原因: Mixing up polygon names (e.g. hexagon = 6 sides, not 5) leads to incorrect angle sum calculations
  - 正确做法: Memorise side counts for common polygons: pentagon=5, hexagon=6, octagon=8
- **错误做法:** Thinking the sum of exterior angles changes with number of sides
  - 原因: Students often assume more sides = larger exterior angle sum, but it is always fixed at 360° for any polygon
  - 正确做法: Recall that exterior angle sum is 360°, regardless of the number of sides of the polygon
- **错误做法:** Claiming a square is not a type of rhombus
  - 原因: Students incorrectly assume rhombuses can only have non-right angles, but squares meet all rhombus definition criteria
  - 正确做法: Remember squares are a special case of both rhombuses and rectangles
- **错误做法:** Using 180° sum for quadrilaterals instead of 360°
  - 原因: Mixing up triangle and quadrilateral interior angle sums leads to missing angle errors
  - 正确做法: Memorise quadrilateral angle sum is 360°, or derive it using the general formula for $n=4$

## 速查表

| Concept | Rule/Formula |
| --- | --- |
| Sum of interior angles (n sides) | $(n-2) \times 180^\circ$ |
| Sum of exterior angles (any polygon) | $360^\circ$ |
| Regular n-gon exterior angle | $\frac{360^\circ}{n}$ |
| Regular n-gon interior angle | $180^\circ - \frac{360^\circ}{n}$ |
| Quadrilateral interior angle sum | $360^\circ$ |
| Congruent polygons | Same shape and size, all corresponding sides/angles equal |

## 下一步

Now that you have mastered polygon properties and angle calculations, you are ready to move on to more advanced geometry topics in the Edexcel IGCSE Mathematics A specification. Next, you will apply these angle rules to problems involving circles, congruent triangles, and trigonometry, which make up a large portion of the geometry and trigonometry unit. You should also practice applying polygon properties to multi-step geometry problems, as these are frequently tested in both foundation and higher tier papers. Make sure you memorise all the rules and formulae in this guide, as none are provided on the exam formula sheet. Regular practice of past paper questions will help you avoid common errors and build speed when solving angle problems.

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