📚 Bearings and Scale Drawings | 方位角与比例尺绘图
Bearings and scale drawings are essential tools for describing direction and distance in practical problems. In Cambridge KS3 Mathematics, you are expected to measure, state and use three-figure bearings, and to interpret or construct accurate scale drawings. This article explains the key ideas, worked examples and common errors so you can approach any question with confidence.
方位角和比例尺绘图是在实际问题中描述方向和距离的重要工具。在剑桥 KS3 数学中，你需要测量、书写和使用三位数方位角，并能够解读或绘制准确的比例尺图。本文讲解核心概念、例题和常见错误，帮助你自信应对相关题目。
1. What Are Three-Figure Bearings? | 什么是三位数方位角？
A bearing is a way of describing direction. It is always measured from north in a clockwise direction and written as a three-digit angle. For example, east has a bearing of 090°, south has a bearing of 180°, west has a bearing of 270°, and north is written as 000° or 360°.
方位角是描述方向的一种方式。它总是从北方开始按顺时针方向测量，并写成三位数的角度。例如，东方的方位角为 090°，南方为 180°，西方为 270°，北方可写成 000° 或 360°。
- Always start at north. 始终从正北开始。
- Measure clockwise. 按顺时针方向测量。
- Write the angle with three digits, such as 045°, not 45°. 角度必须写成三位数，如 045°，而不是 45°。
Bearing = angle from north, measured clockwise
方位角 = 从正北开始顺时针测得的角度
2. Measuring Bearings from North | 从北方测量方位角
To find the bearing of B from A, place the protractor with 0° pointing north at point A. Measure the angle clockwise until you reach the line AB. Always give the answer as three digits. If the angle is less than 100°, add a leading zero to make it a three-figure bearing.
要找出从 A 到 B 的方位角，把量角器的 0° 对准 A 点的北方。沿顺时针方向测量到直线 AB 的角度。答案必须写成三位数，如果角度小于 100°，要在前面加 0，构成三位数方位角。
Example: If the clockwise angle is 65°, write the bearing as 065°. This is not the same as a normal angle of 65°; the leading zero shows that it is a bearing.
例如：如果顺时针量得的角度为 65°，方位角应写作 065°。这与普通的 65° 角不同，前导零表示它是一个方位角。
3. Back Bearings and Opposite Directions | 反向方位角与相反方向
The back bearing is the bearing of A from B, which is the opposite direction of the bearing from B to A. To find a back bearing, add 180° if the original bearing is less than 180°, or subtract 180° if it is 180° or more.
反向方位角是从 B 到 A 的方位角，它是从 A 到 B 的相反方向。求反向方位角的方法是：若原方位角小于 180°，则加上 180°；若原方位角大于或等于 180°，则减去 180°。
Example 1: Bearing of B from A = 075°. Back bearing of A from B = 075° + 180° = 255°.
例 1：从 A 到 B 的方位角为 075°。从 B 到 A 的反向方位角 = 075° + 180° = 255°。
Example 2: Bearing of B from A = 230°. Back bearing = 230° − 180° = 050°.
例 2：从 A 到 B 的方位角为 230°。反向方位角 = 230° − 180° = 050°。
4. Bearings in Real-World Contexts | 实际情境中的方位角
Bearings are used by ships, aircraft and walkers to follow a route. A bearing describes the direction of travel from one point to another. For example, a lighthouse may be on a bearing of 140° from a boat. The boat is then on the back bearing of 320° from the lighthouse.
船舶、飞机和徒步者使用方位角来沿路线行进。方位角描述了从一个点到另一个点的行进方向。例如，灯塔在船的方位角 140° 方向上，那么船就在灯塔的反向方位角 320° 方向上。
- North N = 000° 北方 N = 000°
- East E = 090° 东方 E = 090°
- South S = 180° 南方 S = 180°
- West W = 270° 西方 W = 270°
Always check which point is the starting point. The phrase 'the bearing of A from B' means you stand at B and look towards A.
始终要判断哪一个点是起点。“A 从 B 的方位角”表示你站在 B 点，看向 A 点。
5. Introduction to Scale Drawings | 比例尺绘图简介
A scale drawing is a drawing that represents a real object or area with all lengths reduced or enlarged by the same ratio. The scale tells you how many units on the drawing correspond to real-life units. Scales can be written as a ratio, such as 1:50, or as a statement, such as 1 cm represents 5 m.
比例尺绘图是按照相同比例缩小或放大实际物体或区域的绘图。比例尺说明图上多少单位对应实际生活中的单位。比例可以写成比，如 1:50；也可以写成文字，如 1 厘米代表 5 米。
A scale of 1:50 means the real object is 50 times larger than the drawing. A scale of 2:1 means the drawing is twice as large as the real object, which is useful for enlarging small objects.
比例 1:50 表示实际物体是图上的 50 倍。比例 2:1 表示图上的尺寸是实际物体的两倍，这种比例常用于放大微小物体。
6. Choosing and Using a Scale | 选择和使用比例尺
To choose a scale, first look at the real length and the size of the drawing area. Divide the largest real length by the largest available drawing length to suggest a sensible scale. A scale of 1:n means 1 unit on the drawing represents n units in real life.
选择比例尺时，先看实际长度和图纸可用区域的大小。用最大实际长度除以图纸上可用的最大长度，得出一个合理的比例。比例 1:n 表示图上 1 个单位代表实际 n 个单位。
Drawing length = Actual length ÷ n
图上长度 = 实际长度 ÷ n
Actual length = Drawing length × n
实际长度 = 图上长度 × n
If the scale is 1:100, a real length of 800 cm is drawn as 8 cm. A drawing length of 3 cm represents a real length of 300 cm, or 3 m.
如果比例尺为 1:100，实际长度 800 厘米应画成 8 厘米。图上的 3 厘米代表实际长度 300 厘米，即 3 米。
7. Converting Real Distances and Map Distances | 实际距离与图上距离的换算
ents 5 × 2 = 10 m in real life. A real distance of 15 m is represented by 15 ÷ 2 = 7.5 cm.
知道比例尺后，就可以在图上测量和实际测量之间进行换算。若 1 厘米代表 2 米，图上 5 厘米代表实际 5 × 2 = 10 米。实际距离 15 米在图上应为 15 ÷ 2 = 7.5 厘米。
| Scale 比例尺 | Drawing length 图上长度 | Real length 实际长度 |
| 1 cm : 5 m | 3 cm | 15 m |
| 1 : 1000 | 4 cm | 4000 cm = 40 m |
| 1 : 25000 | 8 cm | 200000 cm = 2 km |
Always convert units carefully. It is often easiest to convert everything to the same unit before multiplying or dividing.
换算单位时要格外仔细。通常先统一单位，再进行乘法或除法计算较为简便。
8. Constructing a Scale Drawing |
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