📚 Bearings and Scale Drawings: KS3 Cambridge Problem Solving | 方位角与比例尺绘图：剑桥KS3解题方法
Bearings and scale drawings are key practical skills in the Cambridge KS3 mathematics curriculum. They combine angle measurement, ratio, unit conversion and accurate construction. This article explains the core rules and works through the type of question often seen around page 232 of Cambridge KS3 materials.
方位角和比例尺绘图是剑桥初中数学课程中重要的实用技能。它们结合了角度测量、比、单位换算和精确作图。本文解释核心规则，并讲解第232页常见类型的问题。
1. What Are Bearings? | 什么是方位角？
A bearing is a three-digit angle used to describe direction. It is always measured clockwise from the North line, so it removes the ambiguity of phrases such as 'turn left' or 'turn right'.
方位角是用来描述方向的三位数角。它总是从正北方向顺时针测量，因此消除了“向左转”或“向右转”这类说法的歧义。
We write bearings with three digits. For example, an angle of 65° is written as 065°, and 5° is written as 005°.
方位角要写成三位数。例如，65° 写作 065°，5° 写作 005°。
| Direction | 方向 | Bearing | 方位角 |
|---|---|
| North | 正北 | 000° |
| East | 正东 | 090° |
| South | 正南 | 180° |
| West | 正西 | 270° |
2. The Three Key Rules of Bearings | 方位角的三条关键规则
Rule 1: Bearings are measured from the North line. The starting line is always North, never East, South or West.
规则一：方位角从正北线开始测量。起始线总是正北线，而不是东、南或西。
Rule 2: Bearings are measured clockwise. This means 090° points exactly to the right, and 180° points directly downwards.
规则二：方位角沿顺时针方向测量。也就是说，090° 指向正右方，180° 指向正下方。
Rule 3: Bearings are written with three digits. Always use leading zeros when the angle is less than 100°.
规则三：方位角写成三位数。当角度小于 100° 时，一定要在前面补零。
3. Measuring Bearings with a Protractor | 使用量角器测量方位角
Place the centre of the protractor at the starting point and align the 0° mark with the North direction. Then read the angle clockwise around the protractor.
将量角器的中心放在起点，并将 0° 刻度与正北方向对齐。然后沿量角器顺时针读取角度。
If the direction is to the right of North, you can read the acute angle directly. For example, a point 40° east of North has a bearing of 040°.
如果方向在正北的右侧，可以直接读取锐角。例如，位于正北以东 40° 的点的方位角是 040°。
h and subtract it from 360°. A point 30° west of North has a bearing of 360° − 30° = 330°.
如果方向在正北的左侧，则从正北逆时针测量角度，并用 360° 减去它。正北以西 30° 的点的方位角是 360° − 30° = 330°。
4. Compass Directions and Bearings | 罗盘方向与方位角
The four cardinal compass directions have fixed bearings: North is 000°, East is 090°, South is 180° and West is 270°.
四个基本罗盘方向有固定的方位角：正北是 000°，正东是 090°，正南是 180°，正西是 270°。
The four intercardinal directions are halfway between the cardinal directions. Northeast is 045°, Southeast is 135°, Southwest is 225° and Northwest is 315°.
四个中间罗盘方向位于基本方向之间。东北是 045°，东南是 135°，西南是 225°，西北是 315°。
ection, divide the bearing by 45°. For example, 270° ÷ 45° = 6, which corresponds to West.
要把方位角转换回罗盘方向，可以用方位角除以 45°。例如，270° ÷ 45° = 6，对应正西。
5. Scale Drawings and Map Ratios | 比例尺绘图与地图比例
A scale drawing shows a real object with every length multiplied by the same scale factor. A map scale such as 1 : 50 000 means that 1 cm on the map represents 50 000 cm in real life.
比例尺绘图是指用同一个比例因子乘以实际物体的所有长度。地图比例 1 : 50 000 表示地图上的 1 cm 代表实际中的 50 000 cm。
al in scale problems. Remember that 100 cm = 1 m and 1000 m = 1 km, so 50 000 cm is 500 m.
单位换算在比例尺问题中非常重要。记住 100 cm = 1 m，1000 m = 1 km，因此 50 000 cm 就是 500 m。
actual length = drawing length × scale factor
实际长度 = 图上长度 × 比例尺因子
th is 7.2 cm on a 1 : 25 000 map, the real distance is 7.2 × 25 000 = 180 000 cm = 1.8 km.
例如，如果一条路径在 1 : 25 000 的地图上是 7.2 cm，那么实际距离是 7.2 × 25 000 = 180 000 cm = 1.8 km。
6. Constructing a Scale Drawing | 绘制比例尺图
Start by choosing a simple scale, such as 1 cm : 10 m or 1 cm : 2 km. Write down the scale and convert every real length into a drawing length before you start.
首先选择一个简单的比例尺，例如 1 cm : 10 m 或 1 cm : 2 km。写下比例尺，并在开始前将每个实际长度转换为图上长度。
Use a ruler and a sharp pencil to draw each line accurately. Draw a fresh North line at each new starting point, because bearings are always measured from North.
使用直尺和削尖的铅笔，准确地画出每一条线。在每个新的起点绘制新的正北线，因为方位角总是从正北测量。
Label every bearing and distance on your diagram. A clear diagram is much easier to check and mark in an exam.
在图上标出每一个方位角和距离。清晰的图在考试中更容易检查和获得分数。
7. Finding Distances from a Scale Drawing | 从比例尺图中求实际距离
Measure the required length on the drawing with a ruler. Then multiply the measured drawing length by the scale factor to obtain the real length.
用直尺测量图上的所需长度。然后将测得的图上长度乘以比例尺因子，得到实际长度。
Always convert the final answer into the unit asked for. If the answer is in cm, convert to m or km when the question requires it.
最终答案一定要转换成题目要求的单位。如果答案是 cm，在题目要求时转换为 m 或 km。
es are 11.5 cm apart on the map. The real distance is 11.5 × 40 000 = 460 000 cm = 4.6 km.
示例：一张地图的比例尺是 1 : 40 000。两个村庄在地图上相距 11.5 cm。实际距离是 11.5 × 40 000 = 460 000 cm = 4.6 km。
8. Combining Bearings and Scale Drawings | 结合方位角与比例尺绘图
Many Cambridge KS3 problems ask you to represent a journey. You need to draw each leg using its bearing and scaled distance, and then find the return distance and bearing.
许多剑桥 KS3 题目要求你表示一段旅程。你需要使用每条线段的方位角和比例距离进行绘制，然后求出返回距离和方位角。
Worked example: A boat sails 6 km on a bearing of 070°, then 8 km on a bearing of 150°. Use the scale 1 cm : 1 km to draw the journey.
解题示例：一艘船以 070° 的方位角航行 6 km，然后以 150° 的方位角航行 8 km。使用比例尺 1 cm : 1 km 绘制这段旅程。
Draw a North line at the start and mark 070° clockwise. Draw the first leg 6 cm long. At the end of that leg, draw a new North line, then mark 150° clockwise and draw the second leg 8 cm long.
在起点画正北线，并顺时针标出 070°。画出第一条长 6 cm 的线段。在该线段末端画一条新的正北线，然后顺时针标出 150°，画出第二条长 8 cm 的线段。
Finally, join the final point back to the start. Measure this final line to find the return distance, and measure the bearing from the final point to the start.
最后，将终点与起点连接起来。测量这条最终线段以求出返回距离，并测量从终点到起点的方位角。
9. Common Mistakes | 常见错误
Do not measure bearings anticlockwise. Even if the angle is acute, the bearing must be the full clockwise turn from North.
不要逆时针测量方位角。即使该角是锐角，方位角也必须是自正北起顺时针转过的完整角度。
Do not forget the three-digit rule. Writing 45° instead of 045° loses a mark in many Cambridge tests.
不要忘记三位数规则。在许多剑桥考试中，写 45° 而不写 045° 会丢分。
Do not ignore units. A scale factor of 1 : 50 000 is in cm, so the result must be converted to m or km when needed.
不要忽视单位。比例尺因子 1 : 50 000 的单位是 cm，因此在需要时必须将结果转换为 m 或 km。
When measuring a return bearing, draw a fresh North line at the new point. Using the original North line will give the wrong answer.
测量返回方位角时，要在新的点绘制新的正北线。使用原来的正北线会得到错误的答案。
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