📚 Bearings and Scale Drawings | 方位角与比例尺绘图
In KS3 Cambridge Mathematics, one of the most practical topics is Bearings and Scale Drawings. This article clarifies how to measure, draw and use bearings to describe direction, and how to interpret scale drawings to find real‑life distances. The skills covered here are essential for map reading, navigation and many GCSE problems.
在 KS3 剑桥数学中，方位角与比例尺绘图是最实用的课题之一。本文将讲解如何测量、绘制和使用方位角来描述方向，以及如何解读比例尺绘图以求出实际距离。这些技能对读图、导航以及许多 GCSE 问题都至关重要。
1. What is a Bearing? | 什么是方位角？
A bearing is a way of describing a direction. It is measured in degrees (°), always starting from North, and moving clockwise. Bearings are always written as three‑figure numbers, so 5° is written as 005°, 45° as 045°, and 315° as 315°. This standard avoids confusion when giving directions between two points.
方位角是描述方向的一种方式。它以度（°）为单位，始终从正北开始，沿顺时针方向测量。方位角总是写成三位数，因此 5° 写成 005°，45° 写成 045°，315° 写成 315°。这一规范可以避免在两点间指路时产生混淆。
2. Measuring Bearings with a Protractor | 用量角器测量方位角
To measure the bearing of a point B from point A, draw a North line at A. Place a protractor with the 0° mark aligned to North, facing clockwise. Read the angle from North clockwise to the line AB. Always check if you need the angle as three figures. If the angle is less than 100°, add leading zeros (e.g. 060°).
要测量点 B 相对于点 A 的方位角，先在 A 处画一条正北线。将量角器的 0° 标记对准正北，顺时针方向放置。读取从正北顺时针旋转到 AB 线的角度。务必确认角度是否为三位数。如果角度小于 100°，需在前面补零（如 060°）。
3. Drawing a Bearing | 绘制方位角
To draw a bearing, start with a marked point and draw a thin vertical North line through it. Place the centre of the protractor on the point, and make a small mark at the required bearing angle measured clockwise from North. Connect the point to this mark with a ruler to show the direction. Label the bearing and distance if a scale is used.
绘制方位角时，先从标记的点画一条细的垂直北线。将量角器的中心对准该点，并在从正北顺时针量取所需方位角的刻度处做一个小标记。用直尺将该点与此标记连接，显示方向。如果使用比例尺，请标注方位角和距离。
4. Reverse Bearings | 反向方位角
bearing of A from B is 070° + 180° = 250°. If the bearing is 210°, reverse bearing is 210° – 180° = 030°.
从 B 返回 A 的方位角称为反向方位角。如果原方位角小于 180°，则加 180°；如果大于 180°，则减 180°。例如，B 相对于 A 的方位角是 070°，则 A 相对于 B 的方位角是 070° + 180° = 250°。若方位角为 210°，反向方位角为 210° – 180° = 030°。
5. The Compass and Bearings | 罗盘与方位角
The main compass points correspond to specific bearings: North is 000°, East is 090°, South is 180° and West is 270°. In KS3 problems, you may be asked to convert a compass direction like NE (045°) or SW (225°) into a bearing. Knowing these conversions speeds up map work.
主要罗盘方向对应特定的方位角：北是 000°，东是 090°，南是 180°，西是 270°。在 KS3 的题目中，可能会要求你将东北（045°）或西南（225°）等罗盘方向转换为方位角。熟悉这些转换可以加快读图速度。
6. Introduction to Scale Drawings | 比例尺绘图简介
A scale drawing represents a real object or distance in a reduced or enlarged size. The scale is given as a ratio, such as 1 : 100, meaning 1 cm on the drawing represents 100 cm (1 m) in real life. Scales can also be given as a statement, like ‘1 cm represents 5 km’.
比例尺绘图用缩小或放大的尺寸表示真实物体或距离。比例尺以比的形式给出，如 1 : 100，表示图上 1 cm 代表实际距离 100 cm（1 m）。比例尺也可以陈述为“1 cm 代表 5 km”这样的形式。
7. Using Scales to Find Real Lengths | 使用比例尺求实际长度
cm long, the real length is 3.5 × 200 = 700 cm, or 7 m. Always convert to sensible units.
要计算实际长度，先用直尺量出图上的距离，再乘以比例因子。如果比例尺是 1 : 200，图形长度是 3.5 cm，那么实际长度就是 3.5 × 200 = 700 cm，即 7 m。务必换算成合理的单位。
8. Drawing a Scale Diagram | 绘制比例图
diagram. Use a ruler and sharp pencil to draw lines accurate to the nearest millimetre. Include the scale and a North line if the diagram involves bearings.
先用给定的比例尺把实际距离换算成图上长度。若比例尺是 1 : 50，实际距离 4 m 转换为 400 cm ÷ 50 = 8 cm。使用直尺和削尖的铅笔，精确到毫米画线。如果图形涉及方位角，记得添加比例尺和北线。
9. Maps and Bearings Together | 地图与方位角的结合
Many KS3 exam questions combine bearings and scale drawings. You may be asked to find the bearing of a town from an airport, then calculate the straight‑line distance between them using the map scale. The key is to draw a clean diagram, measure the bearing carefully, and apply the scale accurately.
许多 KS3 考题都会结合方位角和比例尺绘图。你可能会被要求找出小镇相对于机场的方位角，再使用地图比例尺计算两者间的直线距离。关键在于绘制清晰的示意图，仔细测量方位角，并准确应用比例尺。
10. Common Pitfalls and Tips | 常见错误与技巧
Always start your protractor from North, not from the vertical line of the page unless it is North. Use three‑figure bearings even for answers like 90° (write 090°). When calculating scale, check whether you are converting from drawing to real life or the opposite. Write the scale ratio with the drawing size first. Finally, round real‑life distances sensibly, and always include units.
量角器总是从正北开始，而不是随便找一条垂直线，除非它就是北线。即使答案是 90°，也要写成 090°。计算比例时，要分清是从图上换算到实际还是相反。写比例尺时，图纸尺寸放在前。最后，对实际长度合理取整，并始终带上单位。
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