📚 Bearings and Scale Drawings | 方位角与比例绘图
Bearings describe direction using angles measured clockwise from north. Scale drawings use a ratio to represent real objects at a smaller or larger size on paper. In the Cambridge KS3 mathematics course, you need to measure, draw and interpret bearings and use map scales to find real distances and areas.
方位角使用从正北顺时针测量的角度来表示方向。比例绘图利用比例在纸上以缩小或放大的方式表示真实物体。在剑桥 KS3 数学课程中，你需要测量、绘制和解读方位角，并利用地图比例尺求出实际距离和面积。
1. What Are Bearings? | 什么是方位角
A bearing is a direction measured from the north line in a clockwise direction. It always starts at north, which is 0°, and turns clockwise until it reaches the line of travel. Bearings are used in navigation, map work and construction problems.
方位角是从正北线沿顺时针方向测量的方向。它始终从正北 0° 开始，顺时针旋转到行进方向所在的直线。方位角用于导航、地图作业和几何作图问题。
The angle is always given as a three-figure number. For example, a bearing of 60° is written as 060°, and a bearing of 8° is written as 008°. This rule avoids confusion between directions such as 2° and 20°.
方位角始终写成三位数。例如，60° 的方位角写作 060°，8° 的方位角写作 008°。这条规则可以避免 2° 和 20° 等方向之间的混淆。
- Bearings are measured clockwise from north.
- 方位角从正北开始顺时针测量。
- A bearing is between 000° and 360°.
- 方位角在 000° 到 360° 之间。
2. Writing Bearings with Three Digits | 用三位数表示方位角
Every bearing must contain three digits, so small angles need zeros in front. North is 000°, east is 090°, south is 180° and west is 270°. These four compass directions are important reference points.
每个方位角都必须包含三位数字，所以较小的角度需要在前方补零。正北是 000°，正东是 090°，正南是 180°，正西是 270°。这四个罗盘方向是重要的参考点。
If a bearing is given as 245°, it means the direction lies between south-west and west. The first digit 2 shows the direction is in the third quarter of the compass, while the rest of the angle gives its precise position.
如果方位角为 245°，则表示该方向位于西南和西之间。首位数字 2 表示方向在罗盘的第三象限，其余角度则给出它的精确位置。
| Compass direction | Bearing | 中文方向 |
|---|---|---|
| North | 000° | 正北 |
| East | 090° | 正东 |
| South | 180° | 正南 |
| West | 270° | 正西 |
3. Measuring Bearings with a Protractor | 用量角器测量方位角
To measure a bearing, place the centre of the protractor on the starting point. Align 0° with the north line, then read the angle clockwise from north to the target line. A full-circle protractor is usually easier to use.
要测量方位角，将量角器的中心点放在起点上。将 0° 与正北线对齐，然后从正北顺时针读取到目标线的角度。通常使用全圆量角器会更方便。
If you only have a semicircular protractor, you may need to add 180° when the direction points south of the east-west line. Always check whether the measured angle is acute, obtuse or reflex before writing the final bearing.
如果你只有半圆量角器，当方向指向东西线以南时，可能需要加上 180°。在写出最终方位角之前，始终检查测出的角是锐角、钝角还是优角。
Bearing = clockwise angle from north = 360° − 90° = 270°
4. Drawing Bearings | 绘制方位角
When drawing a bearing, first draw a clear north line at the starting point. Then place a protractor with its centre at the point and mark the required angle clockwise from north. Finally, draw a straight line from the point through the mark.
绘制方位角时，先在起点画出一条清晰的正北线。然后将量角器的中心放在该点上，并从正北顺时针标出所需角度。最后，从该点穿过标记画一条直线。
If the question involves a distance, you must also choose a suitable scale. For example, 1 cm on paper can represent 1 km in real life. Measuring and drawing bearings accurately depends on sharp pencils, correct alignment and careful reading of the protractor.
如果题目涉及距离，你还必须选择合适的比例尺。例如，纸上 1 cm 可以代表实际 1 km。测量和绘制方位角的准确性取决于尖铅笔、正确对齐和认真读取量角器。
5. Back Bearings | 反向方位角
A back bearing is the bearing from the final point back to the starting point. It is found by adding 180° if the original bearing is less than 180°, or subtracting 180° if the original bearing is more than 180°.
反向方位角是从终点返回起点的方位角。如果原方位角小于 180°，则加上 180°；如果原方位角大于 180°，则减去 180°。
°. If the bearing from P to Q is 210°, the back bearing from Q to P is 210° − 180° = 030°.
例如，如果从 A 到 B 的方位角是 065°，那么从 B 到 A 的反向方位角是 065° + 180° = 245°。如果从 P 到 Q 的方位角是 210°，那么从 Q 到 P 的反向方位角是 210° − 180° = 030°。
Back bearing = Bearing ± 180°, keep result between 000° and 360°
6. Scale Drawings and Scale Factors | 比例绘图与比例因子
A scale drawing shows an object with all lengths changed by the same ratio. The scale factor is the number that multiplies a length on the drawing to give the real length. A scale of 1 : 50 means the drawing is 50 times smaller than the real object.
比例绘图以相同的比例改变所有长度。比例因子是纸上长度与真实长度相乘得到实际长度的数。比例 1 : 50 表示图上比实际物体缩小 50 倍。
Write scales in the form 1 : n, where n is the scale factor. If the scale is given as 2 cm : 5 m, convert both sides to the same unit first: 2 cm : 500 cm, which simplifies to 1 : 250.
比例写作 1 : n 的形式，其中 n 是比例因子。如果比例写作 2 cm : 5 m，需要先将两边单位统一：2 cm : 500 cm，化简为 1 : 250。
- Scale 1 : 100 means 1 cm represents 100 cm, or 1 m.
- 比例 1 : 100 表示 1 cm 代表 100 cm，即 1 m。
- Scale 1 : 25 000 means 1 cm represents 25 000 cm, or 250 m.
- 比例 1 : 25 000 表示 1 cm 代表 25 000 cm，即 250 m。
7. Using Map Scales to Find Real Distances | 利用地图比例尺求实际距离
To find a real distance from a map, measure the length on the map in centimetres and multiply it by the scale factor. Then convert the answer into sensible units such as metres or kilometres.
要根据地图求实际距离，先以厘米为单位测量图上的长度，再乘以比例因子。然后将答案转换为合适的单位，例如米或千米。
ance is 7 × 40 000 = 280 000 cm. Since 100 cm = 1 m and 1000 m = 1 km, this equals 2.8 km.
假设地图比例尺为 1 : 40 000，两个城镇在图上相距 7 cm。实际距离为 7 × 40 000 = 280 000 cm。因为 100 cm = 1 m，1000 m = 1 km，所以等于 2.8 km。
Real distance = map distance × scale factor
8. Finding Areas from Scale Drawings | 由比例绘图求面积
Areas do not change by the same scale factor as lengths. If the length scale factor is n, then the area scale factor is n². This means a shape drawn at scale 1 : 10 has lengths divided by 10 and areas divided by 100.
面积的变化比例与长度不同。如果长度比例因子是 n，那么面积比例因子是 n²。这表示以 1 : 10 的比例绘制的图形，长度缩小 10 倍，面积缩小 100 倍。
area is 6 × 5000² = 6 × 25 000 000 = 150 000 000 cm². Dividing by 10 000 gives 15 000 m².
例如，一个公园在比例尺为 1 : 5000 的地图上面积为 6 cm²。实际面积为 6 × 5000² = 6 × 25 000 000 = 150 000 000 cm²。除以 10 000 得到 15 000 m²。
9. Interpreting and Solving Bearing Problems | 解读与求解方位角问题
Bearing problems often describe a journey with two or more legs. You can solve them by drawing a scale diagram using the given bearings and distances. The diagram should include north lines at each point and all three-figure bearings.
方位角问题通常描述包含两段或更多段行程的路线。你可以通过根据给定方位角和距离绘制比例图来解决问题。图中应在每个点画出正北线，并标出所有三位数方位角。
Once the diagram is complete, measure the missing side or angle. For example, a ship sails 8 km on bearing 060°, then 5 km on bearing 150°. Using a scale of 1 cm : 1 km, you can draw the two legs and measure the final distance and direction from the start.
完成图后，测量缺失的边或角。例如，一艘船沿 060° 方位角航行 8 km，再沿 150° 方位角航行 5 km。使用 1 cm : 1 km 的比例，你可以画出两段航程，并测量从起点的最终距离和方向。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
A very common mistake is writing 45° instead of 045°. Bearings must have three figures, so always add leading zeros. Another common error is measuring anticlockwise from north instead of clockwise.
一个常见错误是将 45° 写成 45°，而不是 045°。方位角必须有三位数，所以要始终在前面补零。另一个常见错误是从正北逆时针测量，而不是顺时针测量。
When calculating real distances, do not forget to convert centimetres to metres or kilometres. In area questions, remember to square the scale factor. In diagrams, use a ruler and protractor accurately because small errors in angles can produce large errors in position.
计算实际距离时，不要忘记将厘米转换为米或千米。在面积问题中，记住要将比例因子平方。在图中，要准确使用直尺和量角器，因为角度的微小误差可能造成位置的较大误差。
Three figures, clockwise from north, correct units, n² for area
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