📚 Bearings and Scale Drawings | 方位角与比例图
In KS3 Cambridge Mathematics, bearings and scale drawings connect geometry with real-world navigation. This article revises the key rules, common exam pitfalls, and step-by-step methods for measuring and using bearings in scale diagrams.
在剑桥 KS3 数学中，方位角与比例图将几何与现实中的导航联系起来。本文复习核心规则、常见考试易错点，以及在比例图中测量和使用方位角的逐步方法。
1. What Are Bearings? | 什么是方位角？
A bearing is an angle measured clockwise from the north line. It tells you the direction from one point to another. Bearings are always given as three-figure numbers, such as 045° or 125°.
方位角是从北线起顺时针测量的角度。它表示从一个点到另一个点的方向。方位角总是写成三位数字，例如 045° 或 125°。
The three-figure rule avoids confusion: an angle of 5° is written as 005°, and 60° becomes 060°. This rule keeps bearings clear when solving navigation problems.
三位数字规则可以避免混淆：5° 写为 005°，60° 写为 060°。这一规则使方位角在解决导航问题时保持清晰。
2. Reading and Writing Bearings | 读取与书写方位角
A bearing of 000° points due north, 090° due east, 180° due south and 270° due west. For any angle in between, measure clockwise from the north line.
000° 指向正北，090° 指向正东，180° 指向正南，270° 指向正西。对于任何介于两者之间的角度，都要从北线顺时针测量。
The table below shows common compass directions as three-figure bearings.
下表列出常见罗盘方向对应的三位方位角。
| Compass Direction | Bearing |
|---|---|
| North | 000° |
| North-East | 045° |
| East | 090° |
| South-East | 135° |
| South | 180° |
| South-West | 225° |
| West | 270° |
| North-West | 315° |
3. Compass Directions and Bearings | 罗盘方向与方位角转换
To convert a compass direction to a bearing, first draw the north line at the starting point. Then rotate clockwise until you reach the required direction. For example, north-east lies exactly halfway between north and east, so its bearing is 045°.
要将罗盘方向转换为方位角，先在起点画出北线，然后顺时针旋转到所需方向。例如，东北正好位于北与东的中间，因此它的方位角是 045°。
To convert back, divide the bearing into clockwise quarters. A bearing of 135° is between 090° and 180°, so it points south-east. A bearing of 225° lies between 180° and 270°, so it points south-west.
反过来转换时，把方位角按顺时针分成四个象限。135° 位于 090° 与 180° 之间，因此它指向东南。225° 位于 180° 与 270° 之间，因此它指向西南。
4. Measuring Bearings with a Protractor | 用量角器测量方位角
Place the centre of the protractor on the point where the north line starts. Align the 0° mark with the north line. Read the angle clockwise from north to the line joining the two points.
将量角器的中心放在北线起点上。将 0° 刻度与北线对齐。从北顺时针读到连接两点的线。
If the angle is larger than 180°, measure the smaller anticlockwise angle and subtract it from 360°, or use the outer scale of the protractor. Always check that the final bearing is a three-figure number.
如果角度大于 180°，可先测量较小的逆时针角，再用 360° 减去它，或直接使用量角器的外圈刻度。务必检查最终方位角是否为三位数字。
5. Drawing Bearings from a Point | 从一点绘制方位角
Start by drawing a clear north line at the given point. Use the 0° to 180° line of the protractor as the north line, then mark the required three-figure bearing clockwise.
先在给定点画一条清晰的北线。将量角器的 0° 到 180° 线作为北线，然后顺时针标出所需的三位方位角。
Finally draw a straight line from the point through the mark. Label the bearing in three figures. If you need to show a distance, use the given scale to measure the correct length along this line.
最后从该点过标记画一条直线。用三位数字标注方位角。如果需要表示距离，使用给定比例尺沿这条线量出正确长度。
6. Scale Drawings: What Is a Scale? | 比例图：什么是比例？
A scale drawing represents real objects at a smaller or larger size. The scale tells you how many units on the drawing represent one real unit. For example, a scale of 1 : 50 000 means 1 cm on the map represents 50 000 cm in real life.
比例图以缩小或放大的尺寸表示真实物体。比例尺告诉你图上多少个单位代表实际一个单位。例如，比例尺 1 : 50 000 表示地图上 1 cm 代表实际 50 000 cm。
Scales can be written as ratios, such as 1 : 25 000, or as statements, such as 1 cm represents 0.25 km. Both forms mean the same thing and can be used to convert distances.
比例尺可以写成比的形式，如 1 : 25 000，也可以写成文字说明，如 1 cm 代表 0.25 km。两种形式含义相同，都可以用于换算距离。
7. Converting Real Distances and Map Distances | 实际距离与图上距离的换算
1 : 5000 and a route measures 3.4 cm, the real distance is 3.4 × 5000 = 17 000 cm = 170 m.
将图上距离换算为实际距离，乘以比例尺的倍数。如果地图使用 1 : 5000，一条路线量得 3.4 cm，那么实际距离为 3.4 × 5000 = 17 000 cm = 170 m。
Real distance = map distance × scale factor
实际距离 = 图上距离 × 比例尺倍数
example, 250 m in real life at a scale of 1 : 5000 gives 25 000 ÷ 5000 = 5 cm on the map.
将实际距离换算为图上距离，除以比例尺的倍数。例如，实际 250 m 在 1 : 5000 的比例尺下，图上为 25 000 ÷ 5000 = 5 cm。
8. Using Bearings and Scales Together | 结合使用方位角与比例尺
Many KS3 exam problems combine bearings with scale drawings: first draw the bearing from a point, then measure the required length using the scale.
许多 KS3 考试题将方位角与比例图结合：先从一点绘制方位角，然后按比例尺量出所需长度。
For example, a boat sails 4 km on a bearing of 120°, then 3 km on a bearing of 210°. Use a scale of 1 cm : 1 km to draw the journey, then measure the final bearing and distance from the starting point.
例如，一艘船以方位角 120° 航行 4 km，再以 210° 航行 3 km。使用 1 cm : 1 km 的比例尺画出航程，然后测量终点相对于起点的最终方位角和距离。
Draw the north line at the start, not at the end of the first leg. The final bearing is always measured from the original starting point back to the final position.
在起点处画北线，而不是在第一段航程结束时。最终方位角始终从原始起点到最终位置进行测量。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Do not measure bearings anticlockwise unless the question asks for it. Always use the north line at the starting point, not the ending point.
除非题目要求，否则不要逆时针测量方位角。始终使用起点处的北线，而不是终点处的北线。
If a bearing is less than 100°, keep the leading zero. Write 037° not 37°. Check your scale conversions by asking whether the answer should be larger or smaller.
如果方位角小于 100°，保留前导零。写 037° 而不是 37°。检查比例换算时，判断答案应该变大还是变小。
- Start from north and go clockwise
- Use three figures for every bearing
- Use the correct scale factor for conversions
- Label units clearly after calculating
中文要点：
- 从北线出发，顺时针测量
- 每个方位角都写成三位数字
- 换算时使用正确的比例尺倍数
- 计算后清楚标注单位
10. Exam-Style Practice Questions | 考试风格练习题
Try these questions to test yourself before checking the answers.
先尝试以下题目，再核对答案。
Question 1: Find the bearing of B from A when B is due south-east of A.
问题 1：当 B 在 A 的正东南方向时，求从 A 到 B 的方位角。
Answer: 135°
答案：135°
Question 2: A map has scale 1 : 40 000. Two towns are 6.5 cm apart on the map. Find the real distance in km.
问题 2：一张地图的比例尺为 1 : 40 000。两个城镇在图上的距离为 6.5 cm。求实际距离，单位为 km。
Answer: 6.5 × 40 000 = 260 000 cm = 2.6 km
答案：6.5 × 40 000 = 260 000 cm = 2.6 km
Question 3: A ship sails 5 km north then 5 km east. Find the bearing of the ship from its starting point.
问题 3：一艘船先向北航行 5 km，再向东航行 5 km。求船从起点出发的最终方位角。
Answer: 045°
答案：045°
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