# 相似性

> 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-similarity/

本指南涵盖爱德思IGCSE数学A（4MA1）基础层和高层级的全部相似性相关内容，包括比例尺绘图、长度比例，以及仅高层级要求的相似图形面积/体积比例规则，附带面向考试的完整示例解析。

**先修:** [基础比例计算](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-ratios/); [基础面积和体积公式](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s4-area-volume/)

## 学习目标

- 回忆相似图形的对应角相等、对应边长成比例的性质
- 针对地图、比例尺绘图和相似图形，使用线性比例因子计算未知长度（基础层级要求）
- 对相似图形应用面积比例（$k^2$）和体积比例（$k^3$）规则（高层级要求）
- 求解涉及二维和三维图形相似性的多步考试题型

## 基础层级：相似图形性质与比例尺绘图

**相似图形** — 若两个图形的所有对应角相等，且所有对应边长的比值相同（该比值称为线性比例因子$k$），则这两个图形相似。

*例:* Two equilateral triangles of side length 2cm and 6cm are similar, as all angles are 60° and the side length ratio is 1:3.

所有比例尺绘图和地图都是相似性的实际应用：标注的比例尺给出了绘图/地图与真实物体之间的线性比例因子。例如，1:1000的比例尺代表地图上的1厘米对应真实世界的1000厘米（即10米）。

**例题:** 某地图的比例尺为1:20000，地图上两个城镇之间的距离为3.5厘米。计算两个城镇之间的实际距离，单位为千米。

1. 写出带单位的比例尺：地图上1厘米 = 实际20000厘米
2. $$3.5 \times 20000 = 70000 \text{ cm real}$$
3. 将厘米转换为千米：70000厘米 = 700米 = 0.7千米
4. 最终答案：0.7千米

> **考试提示:** 作答比例尺绘图类题目时务必明确写出单位转换步骤，避免丢失不必要的分数。

## 基础层级：计算相似图形的边长

要计算相似图形中的未知边长，首先找到对应边（等角所对的边），以此计算线性比例因子$k$。如果从小图形放大到大图形，$k>1$；如果从大图形缩小到小图形，$k<1$。

> **tip**
>
> 先标注出所有相等的角，避免混淆对应边，这是长度比例类题目最常见的失分点。

**例题:** 三角形ABC和三角形DEF相似。已知AB=4厘米，DE=12厘米，BC=3厘米，求EF的长度。

1. 识别对应边：AB对应DE，BC对应EF
2. $$k = \frac{DE}{AB} = \frac{12}{4} = 3 \text{ (scaling from ABC to DEF)}$$
3. $$EF = 3 \times 3 = 9 \text{ cm}$$
4. 最终答案：9厘米

> **考试提示:** 始终明确标注比例因子的缩放方向（例如从小图形放大到大图形），避免出现取倒数的错误。

## 高层级：相似图形的面积比例

**面积比例因子** — 对于二维相似图形，若线性比例因子为$k$，则面积比例因子为$k^2$。该规则适用于所有相似图形，包括三角形、圆形和组合图形。

*例:* If linear scale factor is 3, area scale factor is $3^2 = 9$, so the larger shape has 9 times the area of the smaller shape.

**例题:** 两个相似圆形的半径比值为2:5，较小圆形的面积为12平方厘米，计算较大圆形的面积。

1. $$k = \frac{5}{2} = 2.5 \text{ (linear scale factor from small to large)}$$
2. $$Area\ scale\ factor = k^2 = 2.5^2 = 6.25$$
3. $$Area\ of\ larger\ circle = 12 \times 6.25 = 75 \text{ cm}^2$$
4. 最终答案：75平方厘米

> **考试提示:** 绝对不要直接使用线性比例因子计算面积：必须先对其取平方才能得到正确的面积比例。

*计算器:* allowed

## 高层级：相似图形的体积比例

**体积比例因子** — 对于三维相似立体，若线性比例因子为$k$，则体积比例因子为$k^3$。该规则适用于所有三维相似图形，包括立方体、球体、棱柱和棱锥。

*例:* If linear scale factor is 2, volume scale factor is $2^3 = 8$, so the larger solid has 8 times the volume of the smaller solid.

**例题:** 两个相似长方体的线性比例系数为1:4，较大长方体的体积为320 cm³，求较小长方体的体积。

1. $$k = \frac{1}{4} \text{ (linear scale factor from large to small)}$$
2. $$Volume\ scale\ factor = k^3 = \left(\frac{1}{4}\right)^3 = \frac{1}{64}$$
3. $$Volume\ of\ smaller\ cuboid = 320 \times \frac{1}{64} = 5 \text{ cm}^3$$
4. 最终答案：5 cm³

> **考试提示:** 作答体积类题目时请再次确认你已经对线性比例因子取立方，误取平方是这类题目非常高频的错误。

*计算器:* allowed

## 常见错误

- **错误做法:** Mixing up corresponding sides when calculating scale factor
  - 原因: Corresponding sides must be opposite equal angles; using unmatched sides gives an incorrect scale factor
  - 正确做法: Mark equal angles first, then match sides opposite the same sized angles to form your ratio
- **错误做法:** Using linear scale factor directly for area or volume calculations
  - 原因: Area scales with the square of linear ratio, volume with the cube; using linear ratio directly gives the wrong result
  - 正确做法: Square linear scale factor for area questions, cube it for volume questions before applying it
- **错误做法:** Forgetting to convert units correctly in scale drawing questions
  - 原因: Map scales are usually in cm, while real distances are in m or km; unconverted units give answers that are orders of magnitude wrong
  - 正确做法: Write down the scale with units first, then convert your final answer to the required unit explicitly
- **错误做法:** Applying similarity rules to non-similar shapes
  - 原因: Shapes that look similar may not be: all corresponding angles must be equal for shapes to be similar
  - 正确做法: Only use similarity ratio rules if you are told the shapes are similar, or you can confirm all corresponding angles are equal
- **错误做法:** Using the wrong direction of scale factor (e.g. scaling down when you need to scale up)
  - 原因: If you use k from large to small when you need small to large, your answer will be the reciprocal of the correct value
  - 正确做法: State explicitly which shape you are scaling from and to before calculating k, e.g. 'k = large / small' if scaling up

## 速查表

| Tier | Rule Type | Rule |
| --- | --- | --- |
| All Tiers | Similar Shapes Property | Equal corresponding angles, proportional sides |
| All Tiers | Linear Scale Factor | $k = \frac{length\ of\ shape\ 2}{length\ of\ shape\ 1}$ |
| All Tiers | Scale Drawings | 1 unit on drawing = k units real life |
| Higher Only | Area Scale Factor | $Area_{SF} = k^2$ |
| Higher Only | Volume Scale Factor | $Volume_{SF} = k^3$ |

## 下一步

Now that you have mastered similarity rules, you can apply these to a wide range of geometry problems in your Edexcel IGCSE Maths A exam, including enlargement questions, composite shape problems, and multi-step 3D geometry questions. Similarity is often combined with trigonometry and circle theorems in higher tier exam questions, so make sure you revise those topics alongside this one to be prepared for the highest mark questions. Practice past paper questions on similarity to build speed and accuracy, paying close attention to unit conversions and scale factor direction to avoid common mistakes.

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