📚 Understanding Scale Drawings and Maps | 理解比例尺绘图与地图
Scale drawings and maps allow us to represent large real-world objects and distances on a small piece of paper. In the Cambridge Lower Secondary Mathematics Learner’s Book 8, page 151 introduces the idea of using ratios to create and interpret scaled diagrams. This article builds on that foundation, guiding you through calculations involving map scales, similar shapes, and common mistakes to avoid, so you can confidently tackle any scale drawing problem.
比例尺绘图与地图让我们能够将巨大的实物和距离展现在一小张纸上。在剑桥初中数学学生用书第八册第151页，引入了用比率创建和解读比例图的概念。本文以此为基础，带你深入了解地图比例尺、相似图形的相关计算，以及需要避免的常见错误，让你能够自信地解决任何比例尺绘图问题。
1. Introduction to Scale Drawings | 比例尺绘图简介
A scale drawing is a two-dimensional representation of an object or area in which every dimension is reduced or enlarged by the same constant factor. This factor is called the scale factor. For example, a map of a town might use a scale of 1 : 10 000, meaning that 1 cm on the map represents 10 000 cm (or 100 m) in real life.
比例尺绘图是物体或区域的二维表示，其中每一个尺寸都按相同的常数因子缩小或放大。这个因子称为比例因子。例如，一张城镇地图可能采用 1 : 10 000 的比例尺，意味着地图上的 1 cm 代表实际生活中的 10 000 cm（即 100 m）。
The key to working with scale drawings is understanding that all corresponding lengths are proportional. If a building is drawn to scale, the ratio of its height on the drawing to its actual height is the same as the ratio of its width on the drawing to its actual width.
处理比例尺绘图的关键在于理解所有对应长度成比例。如果一座建筑物按比例绘制，那么图上高度与实际高度之比，等于图上宽度与实际宽度之比。
In mathematics, a scale can be expressed as a ratio, a fraction, or in words. The ratio form is the most common in KS3, such as 1 : n, where n is the scale factor. When n is large, the drawing is a reduction; when n is less than 1, the drawing is an enlargement, though we usually deal with reductions in maps.
在数学中，比例尺可以用比率、分数或文字表示。比率形式在 KS3 中最常用，例如 1 : n，其中 n 是比例因子。当 n 很大时，绘图是缩小的；当 n 小于 1 时，绘图是放大的，不过在地图中我们通常处理的是缩小。
2. Reading and Interpreting Scales | 阅读与解释比例尺
Scales on maps and plans can be given in three main ways: as a ratio (1 : 500 000), as a statement (‘1 cm represents 5 km’), or as a scale bar. At KS3, you must be able to convert between these forms and interpret any of them. For instance, a ratio of 1 : 200 000 means 1 cm on the map stands for 200 000 cm in reality.
地图和平面图上的比例尺通常有三种表示方式：以比率表示（1 : 500 000）、以文字说明表示（’1 cm 代表 5 km’）、或以比例尺线段表示。在 KS3 阶段，你必须能够在这些形式之间进行转换，并理解任何一种形式。例如，比率 1 : 200 000 意味着地图上的 1 cm 代表现实中的 200 000 cm。
Converting from a ratio to a statement scale involves changing centimetres to a more convenient unit like metres or kilometres. Since 100 cm = 1 m and 100 000 cm = 1 km, the ratio 1 : 50 000 becomes ‘1 cm represents 0.5 km’ because 50 000 cm = 500 m = 0.5 km.
将比率转换为文字说明需要把厘米换成更方便的单位，如米或千米。因为 100 cm = 1 m，100 000 cm = 1 km，所以比率 1 : 50 000 可以转换为 ‘1 cm 代表 0.5 km’，因为 50 000 cm = 500 m = 0.5 km。
A scale bar provides a visual way to measure distances directly on the map. You can use a ruler to measure a length on the map, then align it with the scale bar to read off the real distance. This is particularly useful when the map is printed or photocopied at a different size, as the ratio may no longer apply, but the bar remains valid.
比例尺线段提供了一种直接在地图上测量距离的直观方法。你可以用尺子量出地图上的长度，然后将其与比例尺线段对齐，读出实际距离。当地图被打印或复印成不同尺寸时，比率可能不再适用，但比例尺线段仍然有效，因此这一点尤其有用。
3. Map Scales as Ratios | 地图比例尺作为比率
Writing a scale as a ratio requires both parts to be in the same unit. Usually, we write it in the simplified form 1 : n. For example, if a football pitch is 90 m long and its plan shows it as 9 cm long, first convert the actual length to cm: 90 m = 9000 cm. The scale is 9 : 9000, which simplifies to 1 : 1000.
将比例尺写成比率要求前后单位相同。通常我们写成简化形式 1 : n。例如，如果一个足球场长 90 m，而其平面图长度为 9 cm，先将实际长度转换为 cm：90 m = 9000 cm。其比例为 9 : 9000，简化为 1 : 1000。
The table below shows common map scales and the real distance represented by 1 cm. This helps you quickly estimate distances without a calculator.
下表展示了几种常见的地图比例尺以及 1 cm 所代表的实际距离。这有助于你无需计算器即可快速估算距离。
| Scale (Ratio) | 1 cm represents (metres) | 1 cm represents (kilometres) |
|---|---|---|
| 1 : 1 000 | 10 m | 0.01 km |
| 1 : 10 000 | 100 m | 0.1 km |
| 1 : 50 000 | 500 m | 0.5 km |
| 1 : 200 000 | 2 000 m | 2 km |
| 1 : 1 000 000 | 10 000 m | 10 km |
从上表可以看出，比例因子越大，比例尺的地图覆盖区域越大，但细节越少。这就是为什么徒步地图使用较小的比例因子（如 1 : 25 000），而国家地图使用较大的比例因子（如 1 : 1 000 000）。
When working with ratios, always remember that the scale factor n tells you how many times larger the real world is compared to the drawing. You will use this n in all your calculations of distance, area, and even volume when dealing with solid models.
当你处理比率时，要记住比例因子 n 告诉你现实世界比绘图大多少倍。在计算距离、面积，甚至在处理立体模型时计算体积时，都会用到这个 n。
4. Calculating Actual Distances from Maps | 根据地图计算实际距离
The core calculation in scale drawings is finding the actual distance given the map distance and the scale. The formula is straightforward: multiply the map distance by the scale factor. For a scale 1 : n,
Actual distance = Map distance × n
比例尺绘图的核心计算是根据地图距离和比例尺求出实际距离。公式很简单：将地图距离乘以比例因子。对于比例尺 1 : n，
实际距离 = 地图距离 × n
For example, on a map with scale 1 : 25 000, a distance of 6.4 cm on the map represents 6.4 × 25 000 = 160 000 cm. Converting to km gives 160 000 ÷ 100 000 = 1.6 km. Always convert the actual distance to a sensible unit: m or km for large distances, cm or mm only for very small ones.
例如，在一张比例尺为 1 : 25 000 的地图上，图上距离为 6.4 cm，代表实际距离 6.4 × 25 000 = 160 000 cm。转换为 km 得到 160 000 ÷ 100 000 = 1.6 km。请务必将实际距离换算为合理的单位：大距离用 m 或 km，只有很小的距离才用 cm 或 mm。
Reverse calculations are also common: given the real distance and the scale, find the map distance. Rearranging the formula gives Map distance = Actual distance ÷ n. If two towns are 18 km apart and the map scale is 1 : 200 000, first express 18 km as 1 800 000 cm, then divide by 200 000 to get 9 cm on the map.
反向计算也很常见：已知实际距离和比例尺，求地图距离。将公式变形可得：地图距离 = 实际距离 ÷ n。如果两个城镇相距 18 km，地图比例尺为 1 : 200 000，首先将 18 km 表示为 1 800 000 cm，然后除以 200 000，得到地图上的距离为 9 cm。
Always be careful with units when performing these calculations. A good habit is to convert all measurements to centimetres before applying the scale factor, because map distances are usually given in cm, and actual distances in ratios are in cm. Doing so prevents costly mistakes in exams.
进行这些计算时，要格外注意单位。一个好习惯是在应用比例因子前将所有测量值都转换为厘米，因为地图距离通常以 cm 给出，而比率中的实际距离也以 cm 为单位。这样做可以避免考试中出现不必要的失分。
5. Drawing to Scale: From Reality to Paper | 按比例绘图：从实际到纸上
When you are asked to draw an object to scale, you need to reduce all real lengths by the same factor. Suppose you need to draw a rectangular garden measuring 15 m by 10 m at a scale of 1 : 200. The first step is to calculate the drawing lengths: 15 m = 1500 cm, so length on paper = 1500 ÷ 200 = 7.5 cm; width = 1000 ÷ 200 = 5 cm.
当要求你按比例绘制物体时，需要将所有实际长度按相同因子缩小。假设你要以 1 : 200 的比例绘制一个长 15 m、宽 10 m 的长方形花园。第一步是计算绘图长度：15 m = 1500 cm，因此纸上长度 = 1500 ÷ 200 = 7.5 cm；宽度 = 1000 ÷ 200 = 5 cm。
Use a ruler and a sharp pencil to draw these scaled lengths accurately. Label the scale on your diagram, and clearly indicate the dimensions either in real units or with a note such as ‘Scale 1 : 200’. This shows the examiner that you understand the scaling process.
使用直尺和削尖的铅笔准确绘制这些缩放后的长度。在图上标注比例尺，并清楚地标出尺寸，无论是使用实际单位还是注明 ‘比例 1 : 200’。这可以向考官展示你理解了缩放过程。
If the shape is irregular, you still reduce each length and angle remains unchanged in a scale drawing. Angles are always preserved under scaling. This is a fundamental property of similarity, which will be explored later. For now, remember that you cannot change angles when scaling—only lengths are multiplied by the scale factor.
如果形状是不规则的，你仍然需要缩小每一条边，而角度在比例尺绘图中保持不变。缩放时角度总是保持不变的。这是相似性的一个基本性质，将在后文详述。目前只需记住，缩放时不能改变角度——只有长度乘以比例因子。
6. Using Scale Factors for Length, Area, and Volume | 使用比例因子处理长度、面积和体积
Scale factors have different effects on length, area, and volume. For a scale factor of n (with n > 1 for a reduction), a length on the drawing is multiplied by n to get the real length. However, an area on the drawing is multiplied by n² to get the real area, and a volume is multiplied by n³.
比例因子对长度、面积和体积的影响各不相同。对于比例因子为 n 的情况（缩小比例时 n > 1），图上的长度乘以 n 得到实际长度。然而，图上的面积需要乘以 n² 才能得到实际面积，而体积则需乘以 n³。
Consider a scale model of a building at 1 : 100. The length scale factor is 100. If a floor area on the model is 5 cm², the real floor area equals 5 × 100² = 5 × 10 000 = 50 000 cm², which is 5 m². If a model volume is 2 cm³, the real volume is 2 × 100³ = 2 × 1 000 000 = 2 000 000 cm³, or 2 m³.
考虑一个比例为 1 : 100 的建筑模型。长度比例因子为 100。如果模型上的一块地板面积为 5 cm²，那么实际地板面积等于 5 × 100² = 5 × 10 000 = 50 000 cm²，也就是 5 m²。如果模型体积为 2 cm³，实际体积是 2 × 100³ = 2 × 1 000 000 = 2 000 000 cm³，即 2 m³。
This concept is often tested in KS3. You might be given two similar shapes and the length ratio, and asked to find the area ratio or volume ratio. For example, if two similar triangles have length ratio 1 : 3, their area ratio is 1 : 9. Understanding this relationship saves time and reduces errors.
这一概念在 KS3 中经常考察。你可能会遇到两个相似图形和它们的长度比例，要求你求面积比或体积比。例如，如果两个相似三角形的长度比为 1 : 3，那么它们的面积比就是 1 : 9。理解这种关系可以节省时间并减少错误。
The table below summarises how to convert from model dimensions to real ones using the scale factor n.
下表总结了如何利用比例因子 n 将模型尺寸转换为实际尺寸。
| Quantity | Conversion formula |
|---|---|
| Length | Real length = Model length × n |
| Area | Real area = Model area × n² |
| Volume | Real volume = Model volume × n³ |
请注意，这种关系仅适用于相似图形。如果两个图形的形状不完全相同，则不能简单使用 n² 或 n³ 的规则。
7. Scale and Similar Shapes | 比例尺与相似形状
Scale drawings are directly linked to the topic of similar shapes. Two figures are similar if one is an enlargement of the other, meaning all corresponding angles are equal and all corresponding sides are in the same ratio. That ratio is the scale factor of the enlargement.
比例尺绘图与相似形状这一主题直接相关。如果一个图形是另一个图形的放大图，即所有对应角相等、所有对应边成相同比例，那么这两个图形就是相似的。这个比例就是放大的比例因子。
On page 151 and beyond, you may see questions where a shape is drawn on a grid and you are asked to redraw it at a scale of 1 : 2 or 2 : 1. This is a direct application of similarity. You must measure each side and multiply by the scale factor, or use the grid to count squares and proportionally adjust.
在第151页及其后的内容中，你可能会看到这样的题目：在一个方格网上画出一个图形，要求你以 1 : 2 或 2 : 1 的比例重新绘制。这是相似性的直接应用。你必须测量每一条边的长度并乘以比例因子，或利用方格网数出格数并按比例调整。
Similar shapes also help you find missing lengths. If you know the scale factor between two similar triangles, you can set up a proportion: side A in figure 1 / side A’ in figure 2 = side B in figure 1 / side B’ in figure 2. This is incredibly useful for indirect measurement, such as finding the height of a tree from its shadow.
相似形状还能帮助你找出缺失的边长。如果已知两个相似三角形之间的比例因子，就可以建立比例关系：图1中边A / 图2中边A’ = 图1中边B / 图2中边B’。这在间接测量中极其有用，例如通过影子来求树的高度。
When working with similarity, always identify the corresponding sides carefully. Mistakes often happen when the orientation of the shape is rotated or reflected. Labelling the vertices and writing down the ratio of corresponding sides helps you keep track.
在处理相似性时，务必仔细识别对应边。当图形经过旋转或反射时很容易出错。给顶点标号并写下对应边的比例，有助于你理清头绪。
8. Word Problems Involving Map Scales | 涉及地图比例尺的应用题
Word problems test your ability to apply scale concepts to real-life situations. A typical problem: ‘On a map of scale 1 : 500 000, the distance between two cities is 8.4 cm. What is the actual distance in kilometres? If a car travels at 70 km/h, how long will the journey take?’
应用题考察你将比例尺概念应用到实际情境中的能力。一个典型的题目是：’在一张比例尺为 1 : 500 000 的地图上，两座城市之间的距离为 8.4 cm。实际距离是多少千米？如果一辆汽车以 70 km/h 的速度行驶，旅途需要多长时间？’
First, find the real distance: 8.4 × 500 000 = 4 200 000 cm = 42 km. Then time = distance ÷ speed = 42 ÷ 70 = 0.6 hours = 36 minutes. These multi-step problems require careful unit conversion and clear working.
首先，求出实际距离：8.4 × 500 000 = 4 200 000 cm = 42 km。然后，时间 = 距离 ÷ 速度 = 42 ÷ 70 = 0.6 小时 = 36 分钟。这类多步骤问题需要仔细的单位换算和清晰的解题步骤。
Another common problem involves area: ‘A park is shown on a plan with scale 1 : 2000. The area of the park on the plan is 15 cm². Calculate the actual area in hectares (1 hectare = 10 000 m²).’ Using the area scale factor: n = 2000, so real area = 15 × 2000² cm² = 15 × 4 000 000 = 60 000 000 cm². Convert to m² by dividing by 10 000: 6000 m² = 0.6 hectares.
另一种常见问题涉及面积：’一个公园在比例尺为 1 : 2000 的平面图上显示面积为 15 cm²。请计算实际面积（以公顷为单位，1 公顷 = 10 000 m²）。’ 利用面积比例因子：n = 2000，因此实际面积 = 15 × 2000² cm² = 15 × 4 000 000 = 60 000 000 cm²。除以 10 000 换算为 m²：6000 m² = 0.6 公顷。
When faced with such problems, underline the given scale, identify what you are solving for (distance, area, time), and write down the relevant formula before plugging in numbers. This structured approach is highly recommended for KS3 exams.
遇到这种问题时，在给定的比例尺下划线，明确你要计算什么（距离、面积、时间），并在代入数字前写下相关公式。这种有条理的方法在 KS3 考试中非常值得推荐。
9. Common Misconceptions and Tips | 常见误区与提示
One frequent error is mixing up the direction of the scale factor. If the scale is 1 : n, remember that the real world is larger, so multiply the map distance by n to get the real distance. Some students incorrectly divide by n, which gives a tiny distance that makes no sense.
一个常见错误是混淆比例因子的运用方向。如果比例尺是 1 : n，请记住现实世界更大，因此要将地图距离乘以 n 才能得到实际距离。有些学生错误地除以 n，得出一个微小的、毫无意义的距离。
Another misconception involves units. Failing to convert all lengths to the same unit before simplifying a scale is a sure way to lose marks. Always write both parts of a scale ratio in the same unit, and when calculating, keep the unit consistent until the final conversion.
另一个误区涉及单位。在化简比例尺前未能将所有长度换算成统一单位，必然会丢分。始终以相同单位书写比例尺比率的两个部分，并在计算时保持单位一致，直到最后再转换。
Regarding area and volume, students often forget to square or cube the scale factor. If a model’s area is known, and you are asked for the real area, check if you used n or n². Sketching a small square of side 1 unit scaled to n units can visually remind you that the area scales by n², not n.
关于面积和体积，学生常常忘记对比例因子进行平方或立方。如果已知模型的面积，要求实际面积，要检查自己用的是 n 还是 n²。画一个边长为 1 单位的小正方形，将其缩放到 n 个单位的正方形，可以直观地提醒你面积是按 n² 缩放的，而不是按 n。
Finally, when drawing to scale, always measure accurately. A slip of a millimetre can change the real distance significantly. Use a sharp pencil, a ruler with millimetre markings, and double-check your work.
最后，按比例绘图时，务必测量准确。一毫米的偏差都可能对实际距离产生显著影响。使用削尖的铅笔、带有毫米刻度的直尺，并仔细检查你的工作。
10. Summary and Key Takeaways | 总结与关键点
Scale drawings and maps are powerful mathematical tools that rely on proportional reasoning. The central idea is the scale ratio 1 : n, which connects any length on the paper to the corresponding real length. Once you master this, many problems become straightforward applications of multiplication or division.
比例尺绘图和地图是依赖比例推理的强大数学工具。其核心思想是比例比率 1 : n，它将纸上的任意长度与对应的实际长度联系起来。一旦你掌握了这一点，许多问题就变成了简单的乘法或除法应用。
Remember the following key points: always work in the same units; use the formula Actual = Map × n; for area, multiply the model area by n²; for volume, multiply by n³. Similar shapes preserve angles and rely on proportional sides. In word problems, write down what you know and what you need to find, plan your steps, and check your unit conversions.
请记住以下关键点：始终使用统一单位；运用公式 实际 = 地图 × n；对于面积，将模型面积乘以 n²；对于体积，乘以 n³。相似形状保持角度不变，并依赖于成比例的边。在应用题中，写下已知条件和需要求解的量，规划你的解题步骤，并检查单位换算。
By practising a variety of questions, including those on page 151 of your textbook and beyond, you will develop fluency and accuracy. Scale drawing skills are not just for the classroom—they are used by architects, engineers, pilots, and anyone who works with maps and plans.
通过练习各种题目，包括教材第151页及之后的内容，你将能流利而准确地运用这些技能。比例尺绘图技能不仅仅用于课堂——建筑师、工程师、飞行员，以及任何使用地图和平面图的人都在运用它们。
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