📚 Bearings and Scale Drawings: Reading and Mapping for Cambridge KS3 | 方位角与比例绘图：剑桥KS3数学精讲
vers bearings and scale drawings, two skills that often appear together in checkpoint-style questions.
本讲对应剑桥KS3练习 p274_2 所考查的内容，讲解方位角与比例绘图这两项常在同一题型中出现的技能。
1. Understanding Bearings | 认识方位角
A bearing is an angle measured clockwise from the north line at a given point. It tells you the direction from one place to another without using left, right, up or down.
方位角是从某一点的北线开始、沿顺时针方向测量的角。它用来表示从一个地点到另一个地点的方向，而不使用左、右、上、下等含糊说法。
Bearings are used in navigation, air traffic control, orienteering and map work. In Cambridge KS3 questions, bearings are always given as three-figure angles.
方位角用于航海、航空交通管制、定向越野和地图工作。在剑桥KS3题目中，方位角始终写成三位数角度。
Bearing of B from A = direction measured clockwise from the north line at A
2. The Three Golden Rules for Bearings | 方位角的三个黄金规则
There are three rules you must follow every time you write or measure a bearing. They are simple but examiners check them carefully.
每次书写或测量方位角时，都需要遵守三条规则。这些规则很简单，但考官会仔细检查。
- Rule 1: Always measure from the north line drawn at the starting point. 中文：务必从起点处画出的北线开始测量。
- Rule 2: Always measure clockwise from the north line. 中文：务必从北线顺时针测量。
- Rule 3: Always write the bearing as three digits, such as 045° or 005°. 中文：务必用三位数书写，例如 045° 或 005°。
A bearing of 80° is written 080°, and a bearing of 8° is written 008°. This avoids confusion in navigation and on diagrams.
80° 的方位角写作 080°，8° 的方位角写作 008°。这可以避免导航和图表中的混淆。
3. Measuring a Bearing with a Protractor | 用量角器测量方位角
To measure the bearing of B from A, first draw a clear north line at point A. Place your protractor so that 0° is on the north line and the centre is at A.
要测量 B 从 A 的方位角，先在 A 点画一条清晰的北线。将量角器放在图上，使 0° 对准北线，中心点对准 A。
Read the angle clockwise from north until you reach the line AB. If the angle is less than 100°, add a leading zero when writing it as a bearing.
从北线开始顺时针读出到达 AB 线的角度。如果角度小于 100°，书写时要在前面补零。
Always draw the north line yourself if it is not given. Use a ruler to keep it vertical and parallel to the map edge.
如果图上没有给出北线，要自己画出。用尺子保持北线竖直并与地图边缘平行。
4. Compass Directions and Common Bearings | 罗盘方向与常见方位角
Compass directions are a quick way to state a bearing. You should know the exact three-figure bearing for each main compass point.
罗盘方向是快速表示方位角的方法。你需要知道每个主要罗盘方向对应的准确三位数方位角。
| Compass Direction | Bearing | 中文方向 |
|---|---|---|
| North | 000° | 北 |
| North-East | 045° | 东北 |
| East | 090° | 东 |
| South-East | 135° | 东南 |
| South | 180° | 南 |
| South-West | 225° | 西南 |
| West | 270° | 西 |
| North-West | 315° | 西北 |
5. Back Bearings and Reverse Directions | 反方位角与相反方向
A back bearing is the bearing you would use to travel in the opposite direction. If you walk from A to B on a bearing of 060°, the back bearing from B to A is not 060°.
反方位角是沿相反方向行走时使用的方位角。如果你从 A 到 B 的方位角是 060°，那么从 B 到 A 的反方位角不是 060°。
To find a back bearing, add 180° if the original bearing is less than 180°, or subtract 180° if it is 180° or greater.
求反方位角时，如果原方位角小于 180°，则加上 180°；如果原方位角大于或等于 180°，则减去 180°。
If θ < 180°, back bearing = θ + 180°; if θ ≥ 180°, back bearing = θ − 180°
Example: bearing 060° → back bearing 240°; bearing 300° → back bearing 120°.
例如：方位角 060° → 反方位角 240°；方位角 300° → 反方位角 120°。
6. Introducing Scale Drawings | 认识比例绘图
A scale drawing is a diagram that represents a real object or distance at a fixed ratio. Bearings are often combined with scale drawings to show a journey or map.
比例绘图是以固定比例表示真实物体或距离的图。方位角常与比例绘图结合，用来表示行程或地图。
A scale is written as 1 : n. This means 1 unit on the drawing represents n units in real life. Both parts of the ratio must use the same unit.
比例尺写作 1 : n。这表示图上的 1 个单位代表实际中的 n 个单位。比例的两部分必须使用相同单位。
presents 50 000 cm in real life. Since 50 000 cm = 500 m = 0.5 km, 1 cm represents 0.5 km.
例如，比例尺 1 : 50 000 表示图上 1 cm 代表实际 50 000 cm。因为 50 000 cm = 500 m = 0.5 km，所以 1 cm 代表 0.5 km。
7. Using Scales: From Drawing to Real Life | 使用比例尺：从图上到实际
To convert a drawing length to a real length, multiply the drawing length by the scale factor n. To convert a real length to a drawing length, divide by n.
将图上长度换算为实际长度时，将图上长度乘以比例因子 n。将实际长度换算为图上长度时，除以 n。
Real length = drawing length × n
000 map, a road measuring 4 cm on the map has real length 4 × 25 000 = 100 000 cm = 1 km.
例如，在 1 : 25 000 地图上，一条路在图上量得 4 cm，实际长度就是 4 × 25 000 = 100 000 cm = 1 km。
nce is 60 m long and the scale is 1 : 200, its drawing length is 60 ÷ 200 = 0.3 m = 30 cm.
如果实际围栏长 60 m，比例尺是 1 : 200，那么图上的长度就是 60 ÷ 200 = 0.3 m = 30 cm。
8. Constructing a Scale Drawing | 制作比例绘图
When a question asks you to construct a scale drawing, follow these steps carefully. Accuracy with ruler and protractor is essential.
当题目要求绘制比例图时，请按照以下步骤仔细操作。准确使用尺子和量角器非常重要。
- Step 1: Choose a suitable scale, such as 1 cm : 5 km. 中文：选择合适的比例尺，例如 1 cm : 5 km。
- Step 2: Convert all real distances into drawing lengths. 中文：把所有实际距离换算成图上长度。
- Step 3: Draw a north line at the starting point. 中文：在起点画出北线。
- Step 4: Use a protractor to measure the bearing and draw a faint guide line. 中文：用量角器测量方位角，并画出浅色辅助线。
- Step 5: Measure the required drawing length along this line and mark the point. 中文：沿这条线量出所需图上长度，并标出点。
- Step 6: If there is a second bearing, draw a new north line at the new point before measuring the next angle. 中文：如果有第二个方位角，先在新点画出新北线，再测量下一个角度。
9. Worked Example: Locating a Ship | 例题：确定船只位置
A ship leaves port P. It sails 15 km on a bearing of 060° to point A, then 10 km on a bearing of 150° to point B. Use a scale of 1 cm : 5 km to draw the journey, then find the distance and bearing of B from P.
一艘船离开港口 P。它沿 060° 方位角航行 15 km 到达 A 点，再沿 150° 方位角航行 10 km 到达 B 点。用 1 cm : 5 km 的比例绘图，并求 B 点相对于 P 的距离和方位角。
→ 2 cm. Draw a north line at P, mark 060° clockwise, and measure a 3 cm line to locate A.
首先换算距离：15 km → 3 cm，10 km → 2 cm。在 P 点画北线，顺时针标出 060°，沿该方向量 3 cm，标出 A 点。
At A, draw a new north line, measure 150° clockwise, and draw a 2 cm line to locate B. Finally, measure PB with a ruler and the bearing of B from P with a protractor.
在 A 点画出新北线，顺时针测量 150°，画出 2 cm 线段标出 B 点。最后用尺子量 PB，用量角器量 B 点相对于 P 的方位角。
6 cm, so the real distance is 3.6 × 5 = 18 km. The measured bearing is approximately 094°.
根据精确绘图，PB 约为 3.6 cm，因此实际距离为 3.6 × 5 = 18 km。测得的方位角约为 094°。
10. Common Mistakes and Exam Tips | 常见错误与考试提示
One of the most common errors is measuring the bearing anticlockwise instead of clockwise. Always start at the north line and move clockwise.
最常见的错误之一是把方位角按逆时针方向测量。务必从北线开始顺时针测量。
Another common mistake is forgetting that a bearing must have three digits. Writing 60° instead of 060° can lose marks even if the angle is correct.
另一个常见错误是忘记方位角必须写成三位数。把 60° 写成 060° 以外的形式，即使角度正确也可能丢分。
When working with scale drawings, always convert units before calculating. If the scale is 1 cm : 5 km, do not mix cm and km without converting.
使用比例绘图时，计算前务必统一单位。如果比例尺是 1 cm : 5 km，不要在未换算的情况下混合使用 cm 和 km。
Finally, show your north lines and construction marks clearly. Examiners award working marks for correct method even if the final measurement is slightly off.
最后，清楚地画出北线和作图痕迹。即使最终测量略有误差，过程方法正确也能获得步骤分。
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