📚 Perimeter and Area of Rectangles and Compound Shapes | 矩形与复合图形的周长和面积
Perimeter and area are two of the most practical measurement skills in the IGCSE Mathematics syllabus. This article explains how to calculate each measure for rectangles, triangles, parallelograms, and compound shapes, with step-by-step methods and exam-style tips.
周长和面积是 IGCSE 数学大纲中最实用的测量技能之一。本文解释如何计算矩形、三角形、平行四边形和复合图形的周长与面积，并提供逐步方法和考试技巧。
1. Key Definitions | 核心定义
The perimeter of a 2D shape is the total distance around its outside edge. It is measured in linear units such as cm, m, or km.
二维图形的周长是围绕其外边缘的总距离。它以线性单位测量，如厘米、米或千米。
The area of a 2D shape is the amount of surface inside its boundary. It is measured in square units such as cm² or m².
二维图形的面积是其边界内部的表面大小。它以平方单位测量，如平方厘米或平方米。
In IGCSE questions, you must show correct units and often convert between metric units before calculating.
在 IGCSE 题目中，你必须写出正确的单位，并且经常需要在计算前转换公制单位。
2. Perimeter of a Rectangle | 矩形的周长
A rectangle has two equal lengths and two equal widths. Its perimeter is found by adding all four sides, or using the formula below.
矩形有两条相等的长和两条相等的宽。它的周长通过四条边相加求得，或使用下面的公式。
P = 2(l + w) or P = 2l + 2w
其中 l 表示长，w 表示宽。
Example: a rectangle has length 8 cm and width 5 cm, so P = 2(8 + 5) = 26 cm.
例如：一个矩形长 8 厘米，宽 5 厘米，因此 P = 2(8 + 5) = 26 厘米。
3. Area of a Rectangle | 矩形的面积
The area of a rectangle is the product of its length and width. This formula is the basis for many other area formulas.
矩形的面积是其长与宽的乘积。这个公式是许多其他面积公式的基础。
A = l × w
always expressed in square units. For example, if l = 8 cm and w = 5 cm, then A = 40 cm².
面积总是用平方单位表示。例如，如果长为 8 厘米、宽为 5 厘米，则面积为 40 平方厘米。
Always check whether the question asks for perimeter or area. Perimeter uses linear units; area uses square units.
务必检查题目要求的是周长还是面积。周长使用线性单位；面积使用平方单位。
4. Units and Conversion | 单位与换算
in the same unit. Common metric conversions are 1 m = 100 cm, 1 cm = 10 mm, 1 km = 1000 m.
在使用公式前，所有长度必须使用相同单位。常用的公制换算有 1 米 = 100 厘米，1 厘米 = 10 毫米，1 千米 = 1000 米。
For area units, the conversion factor is squared: 1 m² = 10,000 cm²; 1 cm² = 100 mm².
对于面积单位，换算系数需要平方：1 平方米 = 10,000 平方厘米；1 平方厘米 = 100 平方毫米。
rt 2.5 m to cm: 2.5 × 100 = 250 cm. Then find the area of a square with side 2.5 m in cm².
例如：将 2.5 米转换为厘米：2.5 × 100 = 250 厘米。然后求边长为 2.5 米的正方形的面积，以平方厘米表示。
5. Area of a Triangle | 三角形的面积
A triangle's area is half the base multiplied by the vertical height. The height must be perpendicular to the base.
三角形的面积是底乘以高的积的一半。高必须垂直于底。
A = ½ × b × h
Example: base = 10 cm, height = 6 cm, so A = ½ × 10 × 6 = 30 cm².
例如：底为 10 厘米，高为 6 厘米，因此面积 = ½ × 10 × 6 = 30 平方厘米。
Do not confuse height with the sloping side. The perpendicular height is the shortest distance from the apex to the base.
不要把高与斜边混淆。垂直高是从顶点到底边的最短距离。
6. Area of a Parallelogram | 平行四边形的面积
A parallelogram is a four-sided shape with opposite sides parallel. Its area is base times vertical height, not base times slant height.
平行四边形是对边平行的四边形。它的面积是底乘垂直高，而不是底乘斜高。
A = b × h
Example: base = 12 cm, vertical height = 7 cm, so A = 12 × 7 = 84 cm².
例如：底为 12 厘米，垂直高为 7 厘米，因此面积 = 12 × 7 = 84 平方厘米。
This formula is closely related to the rectangle: if you cut and slide a triangular piece, the parallelogram becomes a rectangle with the same base and height.
这个公式与矩形密切相关：如果你剪切并平移一个三角形部分，平行四边形就变成具有相同底和高的矩形。
7. Compound Shapes: Splitting Method | 复合图形：分割法
A compound shape is made from two or more simpler shapes. To find its area, split it into rectangles, triangles, or parallelograms, then add the areas together.
复合图形由两个或多个简单图形组成。要求其面积，先将其分割成矩形、三角形或平行四边形，然后将面积相加。
ne rectangle is 5 cm by 3 cm and the other is 7 cm by 3 cm, total area = 15 + 21 = 36 cm².
例如：一个 L 形可以被分割成两个矩形。如果一个矩形为 5 厘米 × 3 厘米，另一个为 7 厘米 × 3 厘米，则总面积为 15 + 21 = 36 平方厘米。
Label each smaller part clearly, and write down each area before adding. This reduces mistakes in multi-step questions.
清楚地标出每个小部分，并在相加前写出每个面积。这样可以减少多步题中的错误。
8. Compound Shapes: Subtraction Method | 复合图形：填补法
Some compound shapes are easier to solve by subtracting a missing area from a larger full rectangle. This is called the subtraction method.
有些复合图形用从一个更大的完整矩形中减去缺失面积的方法更容易求解。这称为填补法。
large 10 cm by 8 cm rectangle has a 4 cm by 3 cm corner cut out. Area = 80 - 12 = 68 cm².
例如：一个大的 10 厘米 × 8 厘米矩形被切掉一个 4 厘米 × 3 厘米的角。面积 = 80 - 12 = 68 平方厘米。
For perimeter, do not simply add the perimeters of the parts. Adding and subtracting lengths must follow the outside edge only.
对于周长，不要简单地相加各部分的周长。边长的加减必须遵循外边缘。
9. Finding Missing Lengths | 找缺失长度
Many IGCSE questions give the area or perimeter and ask for a missing side. Rearrange the formula by inverse operations.
许多 IGCSE 题目给出面积或周长，要求求出一条缺失的边。通过逆运算重新整理公式。
If A = l × w, then w = A ÷ l. If P = 2(l + w), then l + w = P ÷ 2, so l = (P ÷ 2) - w.
如果 A = 长 × 宽，那么宽 = 面积 ÷ 长。如果 P = 2(长 + 宽)，那么 长 + 宽 = 周长 ÷ 2，因此 长 = (周长 ÷ 2) - 宽。
Example: a rectangle has area 72 cm² and length 9 cm. Width = 72 ÷ 9 = 8 cm.
例如：一个矩形的面积为 72 平方厘米，长为 9 厘米。宽 = 72 ÷ 9 = 8 厘米。
Always substitute back into the original formula to check your answer.
务必将答案代回原公式进行检验。
10. Word Problems Strategy | 应用题策略
Read the problem twice and underline the given quantities and the required measurement. Decide whether it is a perimeter or area question.
把题目读两遍，在已知量和要求的测量下划线。判断它是周长问题还是面积问题。
Use keywords: 'fence around a field' suggests perimeter; 'carpet covering a floor' suggests area; 'frame around a picture' suggests perimeter; 'tiles covering a wall' suggests area.
使用关键词：'围栏围绕田野' 提示周长；'地毯覆盖地板' 提示面积；'画框围绕画作' 提示周长；'瓷砖覆盖墙面' 提示面积。
Show every step: convert units, write the formula, substitute numbers, and give a final answer with units.
展示每一步：转换单位、写出公式、代入数字，并给出带单位的最终答案。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: adding the slant side instead of the perpendicular height when finding triangle or parallelogram area.
错误 1：求三角形或平行四边形面积时，用了斜边而不是垂直高。
Mistake 2: writing cm instead of cm² for area. Area is always in square units.
错误 2：面积写成厘米而不是平方厘米。面积总是以平方单位表示。
tting to divide by 2 when finding triangle area. Use the formula A = ½ × b × h every time.
错误 3：求三角形面积时忘记除以 2。每次都使用公式 A = ½ × 底 × 高。
In the exam, draw a sketch and mark all known lengths. For compound shapes, show the lines you used to split the shape.
考试时，画一个草图并标出所有已知长度。对于复合图形，标出你用来分割图形的线。
Remember: perimeter is a length, so units are cm, m, km; area is a surface, so units are cm², m², km².
记住：周长是长度，因此单位是厘米、米、千米；面积是表面，因此单位是平方厘米、平方米、平方千米。
Published by TutorHao | 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