# 二维图形量度

> 爱德思国际GCSE数学A · 4MA1
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s4-mensuration-of-2d-shapes/

本指南涵盖爱德思IGCSE数学A（4MA1）要求的所有核心二维量度技能，包括单位换算、标准图形与组合图形的周长和面积计算、圆形相关内容，以及高阶考纲要求的扇形内容。

**先修:** [基础算术运算与分数乘法](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-arithmetic-fractions/); [基础二维图形（圆、三角形、四边形）的性质相关知识](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s4-basic-2d-shape-properties/)

## 学习目标

- 使用长度换算因子的平方在公制长度单位和面积单位之间进行换算
- 计算由矩形、三角形和圆弓形组成的组合图形的周长
- 识记并应用三角形、矩形、平行四边形、梯形和圆形的面积公式
- 使用角度制方法计算圆、半圆的周长和面积，（仅高阶内容）扇形的周长和面积
- 计算过程中保留全部精度，并按照考题要求对答案进行取整

## 长度与面积的公制单位换算

面积的单位换算使用长度换算因子的平方，因为面积是二维度量。例如，由于1 m = 100 cm，因此1 m² = (100 cm)² = 10,000 cm²。

**面积单位换算规则** — 进行面积单位换算时，先求出对应长度单位换算因子的平方，再根据要求做乘法或除法运算。

**例题:** 将2.5 m²换算为cm²。

1. 步骤1：回忆米与厘米之间的长度换算关系：1 m = 100 cm。
2. $$1 \text{ m}^2 = (100 \text{ cm})^2 = 10,000 \text{ cm}^2$$
3. 步骤2：将给定面积乘以换算因子：$2.5 \times 10,000 = 25,000$。
4. 最终答案：25,000 cm²

> **考试提示:** 请务必反复核对面积单位换算：若换算为更小的单位，数值结果会变大，反之则变小。

*计算器:* allowed

## 二维图形的周长

周长是二维图形边缘的总长度。对于由三角形、矩形、半圆或扇形组成的组合图形，将所有外边缘的长度相加，注意不要重复计算内部边。

**组合图形的周长** — 图形所有外边缘的长度之和，包括来自圆弓形的曲线边缘。

**例题:** 将直径为8 cm的半圆无重叠地拼接在长10 cm、宽8 cm的矩形的8 cm长边上，计算所得图形的周长。

1. 步骤1：列出图形的所有外边缘：矩形的三条边，加上半圆的弧形边。
2. $$3 \text{ rectangle sides} = 10 + 10 + 8 = 28 \text{ cm}$$
3. 步骤2：计算半圆弧长：即直径为8 cm的完整圆的周长的一半。
4. $$Semicircle arc length = \frac{1}{2} \times \pi d = \frac{1}{2} \times \pi \times 8 = 4\pi \approx 12.57 \text{ cm}$$
5. 步骤3：将各段长度相加得到总周长：
6. $$28 + 4\pi \approx 40.57 \text{ cm} (2 dp)$$

> **考试提示:** 周长计算中切勿计入内部边（例如本例中半圆与矩形相接的边）。

*计算器:* allowed

## 标准四边形与三角形的面积

你必须熟记三角形、矩形和平行四边形的面积公式，这些公式不会在考试公式表中给出。梯形面积公式会提供，但你仍需熟练掌握其正确应用方法。

- Area of rectangle = $length \times width$
- Area of triangle = $\frac{1}{2} \times base \times perpendicular height$
- Area of parallelogram = $base \times perpendicular height$
- Area of trapezium = $\frac{1}{2}(a + b)h$, where $a$ and $b$ are the lengths of the two parallel sides, $h$ is the perpendicular distance between them.

**例题:** 计算平行边长分别为6厘米和10厘米、垂直高度为4厘米的梯形的面积。

1. 步骤1：确定梯形面积公式中的各参数值：$a = 6$，$b = 10$，$h = 4$。
2. $$Area = \frac{1}{2}(6 + 10) \times 4$$
3. $$= \frac{1}{2} \times 16 \times 4 = 32 \text{ cm}^2$$

> **考试提示:** 计算三角形、平行四边形和梯形的面积时，请务必使用垂直高度，而非斜边长度。

*计算器:* allowed

## 圆形：周长与面积

考试要求熟记圆形相关公式，这些公式不会印在公式表中。你需要经常计算周长、面积以及半圆（整圆的一半）的对应数值。

**圆形核心公式** — 周长$C = \pi d = 2\pi r$，面积$A = \pi r^2$，其中$r$为半径，$d$为直径，满足$d=2r$。对于半圆，弧长$=\frac{1}{2} \times 2\pi r = \pi r$，面积$=\frac{1}{2} \pi r^2$。

**例题:** 计算半径为5厘米的半圆的面积，结果保留1位小数。

1. 步骤1：先回忆整圆的面积公式，再将结果取半得到半圆面积。
2. $$Area of full circle = \pi r^2 = \pi \times 5^2 = 25\pi$$
3. $$Area of semicircle = \frac{1}{2} \times 25\pi = 12.5\pi \approx 39.3 \text{ cm}^2 (1 dp)$$

> **考试提示:** 如果题目要求给出精确答案，请保留含$\pi$的形式，不要将结果舍入为小数。

*计算器:* allowed

## 仅高阶考试要求：扇形

高阶考生需要使用角度制公式计算圆弧长度和扇形面积（本考试完全不涉及弧度制内容）。扇形是圆形的切片，由以度为单位的圆心角$\theta$定义。

- Arc length of sector = $\frac{\theta}{360} \times 2\pi r$
- Area of sector = $\frac{\theta}{360} \times \pi r^2$
- Perimeter of sector = arc length + $2r$ (the two radii forming the straight edges of the sector)

**例题:** 计算半径为7厘米、圆心角为60度的扇形的周长和面积，结果保留2位小数。

1. 步骤1：先计算弧长，用于后续求周长：
2. $$Arc length = \frac{60}{360} \times 2\pi \times7 = \frac{1}{6} \times 14\pi \approx 7.33 \text{ cm}$$
3. 步骤2：加上两条半径得到总周长：
4. $$Perimeter = 7.33 + 7 +7 = 21.33 \text{ cm} (2 dp)$$
5. 步骤3：计算扇形面积：
6. $$Area = \frac{60}{360} \times \pi \times 7^2 = \frac{1}{6} \times 49\pi \approx 25.66 \text{ cm}^2 (2 dp)$$

> **考试提示:** 计算扇形周长时记得加上两条半径：常见错误是仅将弧长作为扇形的周长。

*计算器:* allowed

## 常见错误

- **错误做法:** Using linear conversion factor for area units, e.g. converting 2 m² to cm² as 2 × 100 = 200 cm².
  - 原因: Area is a 2D measure, so you need to square the linear conversion factor.
  - 正确做法: Multiply by $100^2 = 10,000$, so 2 m² = 20,000 cm².
- **错误做法:** Including internal edges when calculating perimeter of composite shapes.
  - 原因: Perimeter only counts the outer boundary of the shape, not edges that are inside where two shapes join.
  - 正确做法: Trace the outside of the shape with your finger to count only external edges, ignoring any overlapping internal sides.
- **错误做法:** Using slant height instead of perpendicular height for area of triangles, parallelograms or trapezia.
  - 原因: The area formulae are derived using the perpendicular distance between the base and the opposite side/vertex.
  - 正确做法: Always use the height that is at a 90-degree angle to the base of the shape.
- **错误做法:** Forgetting to add the two radii when calculating sector perimeter, only giving the arc length.
  - 原因: The sector is a closed shape, so its perimeter includes the two straight radii edges as well as the curved arc.
  - 正确做法: Calculate arc length first, then add 2r to get the full sector perimeter.
- **错误做法:** Using diameter instead of radius in the circle area formula $A = \pi r^2$.
  - 原因: The area formula uses radius, not diameter.
  - 正确做法: If you are given diameter, divide by 2 first to get the radius before substituting into the area formula.

## 速查表

| Shape Type | Perimeter Formula | Area Formula | Recall or Given? |
| --- | --- | --- | --- |
| Rectangle | $2(l + w)$ | $l \times w$ | Recall |
| Triangle | Sum of 3 sides | $\frac{1}{2} \times b \times h$ | Recall |
| Parallelogram | Sum of 4 sides | $b \times h$ | Recall |
| Trapezium | Sum of 4 sides | $\frac{1}{2}(a + b)h$ | Given |
| Full Circle | $\pi d = 2\pi r$ | $\pi r^2$ | Recall |
| Semicircle | $\pi r + 2r$ | $\frac{1}{2}\pi r^2$ | Recall |
| Sector (Higher) | $\frac{\theta}{360} \times 2\pi r + 2r$ | $\frac{\theta}{360} \times \pi r^2$ | Recall |

## 下一步

Now that you have mastered 2D mensuration, you are ready to move on to 3D mensuration (surface area and volume) which builds directly on the area formulae you learned here. You will also use these perimeter and area skills in other geometry topics including similar shapes, trigonometry and vector problem-solving. Make sure you practice past paper questions for both foundation and higher tier content to reinforce your recall of required formulae and avoid common calculation mistakes. If you are sitting the higher tier exam, ensure you are confident applying sector formulae correctly as these are frequent high-mark question topics.

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来自 [OwlsPrep](https://www.owlsprep.com) —— A-Level / IB / AP / IGCSE 免费学习指南，依据官方考纲编写。原页面：https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s4-mensuration-of-2d-shapes/