📚 Cambridge KS3 Maths: Scale Drawings and Map Scales (p166 Q2 Style) | 剑桥 KS3 数学：比例图与地图比例尺（p166 第2题类型）
Scale drawings and map scales are essential tools in Cambridge KS3 Mathematics. They connect classroom ratio work to real-world maps, architectural plans and scale models. A question such as p166 Q2 typically asks you to interpret a scale, convert a measurement, or draw a scaled version of a shape. This article explains the key methods step by step.
比例图和地图比例尺是剑桥 KS3 数学的重要工具。它们把课堂上的比例知识与真实地图、建筑平面图和比例模型联系起来。像 p166 第2题这类题目通常会要求你解读比例尺、换算测量值，或绘制图形的缩放版本。本文一步步讲解关键方法。
1. What Is a Scale Drawing? | 什么是比例图？
A scale drawing is a copy of a real object or distance in which all lengths are multiplied or divided by the same number. The drawing looks the same in shape, but its size is changed. Maps, model cars, floor plans and technical diagrams are all examples of scale drawings.
比例图是真实物体或距离的复制图，其中所有长度都乘以或除以同一个数。图形的形状不变，但大小改变。地图、汽车模型、房屋平面图和技术图表都是比例图的例子。
For example, a plan might use the scale 1 : 100. This means that every 1 cm on the plan represents 100 cm in real life. The ratio 1 : 100 is the scale factor written as a ratio.
例如，一张平面图可能使用 1 : 100 的比例尺。这表示图上的每 1 cm 代表实际生活中的 100 cm。比值 1 : 100 就是以比值形式书写的比例因子。
scale = drawing length : real length
比例尺 = 图上长度 : 实际长度
2. Reading and Writing Map Scales | 读取和书写地图比例尺
A map scale can be written in three main ways: as a ratio, as a statement, or as a scale bar. In Cambridge KS3 questions you will usually see the ratio form, such as 1 : 25 000 or 1 : 50 000.
地图比例尺可以写成三种主要形式：比值形式、文字说明形式或比例条。在剑桥 KS3 题目中，你通常会看到比值形式，例如 1 : 25 000 或 1 : 50 000。
The ratio 1 : 25 000 means that 1 unit on the map represents 25 000 of the same units on the ground. If you measure in centimetres, then 1 cm on the map represents 25 000 cm in real life. Since 100 cm = 1 m and 1000 m = 1 km, this is the same as 250 m or 0.25 km.
比值 1 : 25 000 表示地图上的 1 个单位代表地面上 25 000 个相同单位。如果以厘米为单位测量，那么地图上的 1 cm 代表实际 25 000 cm。由于 100 cm = 1 m，1000 m = 1 km，这等于 250 m 或 0.25 km。
| Scale ratio | Statement form |
| 1 : 10 000 | 1 cm = 100 m |
| 1 : 25 000 | 1 cm = 250 m = 0.25 km |
| 1 : 50 000 | 1 cm = 500 m = 0.5 km |
Always check whether the question wants the answer in centimetres, metres or kilometres. Converting units carefully is part of the skill being tested.
始终检查题目要求答案使用厘米、米还是千米。仔细换算单位正是考查能力的一部分。
3. Converting Map Distances to Real Distances | 将地图距离转换为实际距离
To find a real distance from a map, multiply the measured map distance by the scale factor. This works because the map has shrunk the real length by that factor.
要根据地图求实际距离，将测量的地图距离乘以比例因子即可。这是因为地图已经按该因子缩小了实际长度。
real distance = map distance × scale factor
实际距离 = 地图距离 × 比例因子
Suppose a map has scale 1 : 50 000 and two villages are 4 cm apart on the map. The real distance is 4 × 50 000 = 200 000 cm. Then convert to kilometres: 200 000 cm = 2000 m = 2 km.
假设一张地图的比例尺是 1 : 50 000，两个村庄在地图上相距 4 cm。实际距离为 4 × 50 000 = 200 000 cm。然后换算为千米：200 000 cm = 2000 m = 2 km。
You can also set up an equivalent ratio. Write 1 : 50 000 = 4 : x, then solve by multiplying 4 × 50 000. Both methods give the same result, and choosing one consistent method helps you avoid errors.
你也可以列出等价比。写为 1 : 50 000 = 4 : x，然后通过 4 × 50 000 求解。两种方法结果相同，选择一种一致的方法有助于避免错误。
4. Converting Real Distances to Map Distances | 将实际距离转换为地图距离
To find how far apart two places should be on a map, divide the real distance by the scale factor. You must first put both quantities in the same unit before dividing.
要求两个地点在地图上应相距多远，将实际距离除以比例因子。在相除之前，必须先把两个量化为相同单位。
map distance = real distance ÷ scale factor
图上距离 = 实际距离 ÷ 比例因子
For example, a road is 3 km long in real life and the map scale is 1 : 25 000. Write 3 km as 300 000 cm. Then map distance = 300 000 ÷ 25 000 = 12 cm.
例如，一条道路实际长 3 km，地图比例尺为 1 : 25 000。将 3 km 写为 300 000 cm。那么图上距离 = 300 000 ÷ 25 000 = 12 cm。
Some questions ask you to draw a line segment of this length, so measure it carefully with a ruler from 0, not from the edge of the ruler.
有些题目会要求你画出这个长度的线段，所以用尺子从 0 刻度开始仔细测量，而不是从尺子边缘开始。
5. Changing Units Correctly | 正确换算单位
Unit conversion is one of the most important sub-skills in scale problems. In Cambridge KS3 Mathematics you need to move confidently between centimetres, metres and kilometres.
单位换算是比例尺问题中最重要的子技能之一。在剑桥 KS3 数学中，你需要熟练地在厘米、米和千米之间转换。
1 cm = 10 mm
1 m = 100 cm
1 km = 1000 m = 100 000 cm
1 cm = 10 mm，1 m = 100 cm，1 km = 1000 m = 100 000 cm。
A useful shortcut for map scales is to remember that 100 000 cm = 1 km. So if you have 200 000 cm, divide by 100 000 to get 2 km. If you have 3 km, multiply by 100 000 to get 300 000 cm.
地图比例尺的一个有用速记是：100 000 cm = 1 km。因此如果有 200 000 cm，除以 100 000 得到 2 km。如果有 3 km，乘以 100 000 得到 300 000 cm。
When a question mixes units, convert to the smaller unit first. This keeps the arithmetic simple and reduces the chance of dividing in the wrong direction.
当题目混用单位时，先换算为较小单位。这样算式更简单，也能减少除错方向的可能。
6. Scale Drawings of Shapes | 图形的比例图
Scale drawing is not only about maps. You may need to enlarge or reduce a rectangle, triangle or irregular shape using a given scale. Every side must be multiplied by the same scale factor.
比例图不仅用于地图。你可能需要按给定比例放大或缩小矩形、三角形或不规则图形。每条边都必须乘以相同的比例因子。
For example, a rectangle measures 5 cm by 3 cm. It is enlarged by scale factor 3. The new dimensions are 15 cm by 9 cm. The shape remains a rectangle because both dimensions changed by the same factor.
例如，一个矩形长 5 cm、宽 3 cm。按比例因子 3 放大后，新尺寸为 15 cm 和 9 cm。由于两个尺寸都按相同因子变化，图形仍然是矩形。
If you are asked to draw a scale version, start by labelling the original lengths. Then multiply each length by the given factor, and measure the new lengths with a ruler. Check that angles stay the same: scaling changes size, not shape.
如果要求你画一个比例版本，先标注原始长度。然后将每个长度乘以给定因子，并用尺子测量新长度。注意角度保持不变：比例缩放改变大小，不改变形状。
7. Bearings on Scale Diagrams | 比例图上的方位角
Some Cambridge KS3 questions combine scale drawing with bearings. A bearing measures direction clockwise from north. Bearings are always written with three digits, such as 045° or 135°.
一些剑桥 KS3 题目将比例图与方位角结合考查。方位角是从北顺时针测量的方向。方位角始终写成三位数，例如 045° 或 135°。
To draw a bearing, place a protractor over the point, align 0° with north, and measure clockwise. Then use the scale to draw the correct distance along that bearing line.
要画方位角，将量角器放在点上，使 0° 对准北方，然后顺时针测量。接着使用比例尺沿着该方位线画出正确的距离。
For example, a ship sails from a harbour on a bearing of 060° for 5 km. If the scale is 1 : 50 000, then 5 km = 500 000 cm, and the map distance is 10 cm. Draw a north line, measure 60° clockwise, and draw a 10 cm line.
例如，一艘船从港口出发，沿 060° 方位航行 5 km。如果比例尺为 1 : 50 000，那么 5 km = 500 000 cm，图上距离为 10 cm。画出指北线，顺时针量出 60°，并画出 10 cm 长的线段。
Questions often ask for the return bearing. To find it, add 180° if the original bearing is less than 180°, or subtract 180° if it is 180° or more.
题目经常要求返航方位角。求法：如果原方位角小于 180°，则加 180°；如果大于或等于 180°，则减 180°。
8. Area Scale Factor | 面积比例因子
When a shape is enlarged by a linear scale factor k, its area is enlarged by k². This is a very common trap in scale questions, especially when converting map areas.
当一个图形按线性比例因子 k 放大时，其面积按 k² 放大。这是比例尺题目中非常常见的陷阱，尤其是在换算地图面积时。
area scale factor = k²
面积比例因子 = k²
Suppose a map scale is 1 : 50 000. The linear scale factor is 50 000. If a lake has area 2 cm² on the map, the real area is 2 × 50 000² = 2 × 2 500 000 000 = 5 000 000 000 cm². Converting to km² gives 0.5 km².
假设地图比例尺为 1 : 50 000。线性比例因子是 50 000。如果一座湖在地图上的面积是 2 cm²，那么实际面积为 2 × 50 000² = 2 × 2 500 000 000 = 5 000 000 000 cm²。换算为 km² 得到 0.5 km²。
Remember: 1 m² = 10 000 cm² and 1 km² = 1 000 000 m². Do not apply the linear factor to an area. If you find yourself multiplying an area by 50 000 instead of 50 000², stop and check.
记住：1 m² = 10 000 cm²，1 km² = 1 000 000 m²。不要将线性因子直接用于面积。如果你发现自己在用 50 000 而不是 50 000² 乘面积，请停下来检查。
9. Common Mistakes | 常见错误
One common mistake is forgetting to convert units before multiplying or dividing. For example, writing 3 km as 3 cm and multiplying by the scale factor gives a nonsense result.
一个常见错误是在乘法或除法之前忘记换算单位。例如把 3 km 写成 3 cm 再乘以比例因子，会得到荒谬的结果。
Another mistake is reading 1 : 25 000 as 1 cm = 25 km. It is 1 cm = 25 000 cm, which is only 0.25 km. Always move from cm to km by dividing by 100 000.
另一个错误是把 1 : 25 000 读成 1 cm = 25 km。实际上它是 1 cm = 25 000 cm，仅为 0.25 km。始终通过除以 100 000 将 cm 换算为 km。
Students also confuse area scale factor with linear scale factor. Use k for lengths and k² for areas. Finally, when drawing bearings, remember to measure clockwise from north, not from the nearest vertical grid line.
学生也容易混淆面积比例因子和线性比例因子。长度使用 k，面积使用 k²。最后，画方位角时记住从北顺时针测量，而不是从最近的竖直线开始测量。
10. Worked Exam-Style Example | 考试题型示例
A map has scale 1 : 25 000. Two towns are 6.4 cm apart on the map. Calculate the actual distance in kilometres.
一张地图的比例尺为 1 : 25 000。两个城镇在地图上相距 6.4 cm。计算实际距离，以 km 为单位。
Real distance = 6.4 × 25 000 = 160 000 cm. Since 100 000 cm = 1 km, 160 000 cm = 1.6 km. The actual distance is 1.6 km.
实际距离 = 6.4 × 25 000 = 160 000 cm。由于 100 000 cm = 1 km，160 000 cm = 1.6 km。实际距离为 1.6 km。
On the same map, a river is 3 km long in real life. Work out the length of the river on the map in centimetres.
在同一张地图上，一条河流实际长 3 km。计算该河流在地图上的长度，以 cm 为单位。
3 km = 300 000 cm. Map distance = 300 000 ÷ 25 000 = 12 cm. A 12 cm line should be drawn on the map.
3 km = 300 000 cm。图上距离 = 300 000 ÷ 25 000 = 12 cm
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