📚 Bearings and Scale Drawings | 方位角和比例尺
Bearings and scale drawings are essential tools used in navigation, map reading, and engineering. A bearing gives the direction of one point from another, measured in degrees clockwise from North. A scale drawing represents real objects at a reduced or enlarged size while keeping the same proportions. In this article, we will explore the rules of bearings, how to measure and draw them, and how to use scale drawings to find real-life distances and directions.
方位角和比例尺是用于导航、地图判读和工程中的重要工具。方位角给出从一点到另一点的方向，以度数从正北顺时针测量。比例尺图按比例缩小或放大表示真实物体，同时保持相同比例。在本文中，我们将探讨方位角的规则，如何测量和绘制它们，以及如何使用比例尺图求现实中的距离和方向。
1. What are Bearings? | 什么是方位角？
A bearing is a way of describing direction precisely. It is an angle measured clockwise from the north direction. Bearings are always written using three digits, so an angle of 50° becomes 050°, and 8° is written as 008°. This three-figure system avoids confusion. When you see a bearing like 120°, you know to turn 120° clockwise from north to face the target.
方位角是精确描述方向的一种方式。它是一个从正北方向顺时针测量的角度。方位角总是用三位数字书写，因此 50° 写作 050°，8° 写作 008°。这种三位数表示法避免了混淆。当你看到一个方位角如 120°，就知道要从正北顺时针转 120° 来面对目标。
2. The Three Rules of Bearings | 方位角的三个规则
There are three essential rules for bearings. First, bearings are always measured from North. Second, they are measured clockwise from North. Third, a bearing is always expressed as a three-digit number using leading zeros where necessary (e.g. 045°). These rules ensure that anyone reading a bearing anywhere in the world interprets it in the same way.
方位角有三个基本规则。第一，方位角总是从正北开始测量。第二，从正北按顺时针方向测量。第三，方位角总是用三位数字表示，必要时在前面补零（例如 045°）。这些规则确保世界上任何地方的人读取方位角时都以相同的方式理解它。
3. Measuring Bearings | 测量方位角
To measure the bearing of point B from point A, place a protractor on the diagram with the 0° mark pointing north. Draw a north line at A, then measure the angle clockwise from North to the line AB. Suppose AB points exactly to the east; the bearing is 090°. If AB points south-west, the bearing is 225° (which is 180° + 45°). On an exam diagram, always use a protractor carefully and read the outer scale for clockwise angles.
要测量从点 A 到点 B 的方位角，将量角器放在图上，0° 标记指向北方。在 A 点画一条正北线，然后从北按顺时针方向测量到线段 AB 的角度。假设 AB 正好指向东方，方位角就是 090°。如果 AB 指向西南，方位角则是 225°（即 180° + 45°）。在考试图里，务必小心使用量角器，并读取外圈刻度来获得顺时针角度。
4. Drawing Bearings | 绘制方位角
To draw a line on a given bearing from a point, first draw a vertical north line through that point. Place your protractor with the centre on the point and the baseline along the north line. Measure the required angle clockwise and mark it. Then draw a straight line through the point and your mark. For example, to draw a bearing of 140°, start at north, rotate clockwise 140°, mark, and draw the line. This technique is used to construct scale drawings of routes.
要从一个点沿给定方位角画线，首先通过该点画一条垂直的北向线。将量角器中心对准该点，底边沿着北向线。顺时针测量所需角度并做标记。然后通过该点和标记画一条直线。例如，要画出方位角 140°，从北开始顺时针转 140°，做标记，然后画线。这项技巧用于构建路线的比例尺图。
5. Back Bearings | 反向方位角
A back bearing is the opposite direction of a given bearing. If you know the bearing from A to B, the bearing from B back to A is the back bearing. To find it: if the original bearing is less than 180°, add 180°; if it is more than 180°, subtract 180°. This is because the reverse direction is a half-turn (180°) away.
反向方位角是指给定方位角的相反方向。如果你知道从 A 到 B 的方位角，那么从 B 返回 A 的方位角就是反向方位角。求法如下：如果原始方位角小于 180°，就加 180°；如果大于 180°，就减 180°。这是因为反向刚好转了半圈（180°）。
Back bearing = Bearing + 180° (if Bearing < 180°) or Bearing − 180° (if Bearing > 180°)
例如，方位角为 045° 时，反向方位角 = 045° + 180° = 225°。方位角为 300° 时，反向方位角 = 300° − 180° = 120°。反向方位角在航行和远足路线规划中非常重要。
6. Bearings and Parallel Lines | 方位角与平行线
When working with bearings on diagrams that include parallel north lines, you can apply angle facts about parallel lines. North lines drawn from different points are parallel. Alternate angles and corresponding angles can help you find unknown bearings. For example, if the bearing of B from A is 070°, and you draw the north line at B parallel to the one at A, the interior angle you measure may equal 70° in certain configurations.
当图中包含平行的北向线时，可以利用平行线的角度性质来求方位角。从不同点画出的北向线是平行的。内错角和同位角可以帮助你求出未知的方位角。例如，若从 A 到 B 的方位角为 070°，而在 B 点画的北向线与 A 点的北向线平行，某些情况下你测得的内角也会是 70°。
7. Introduction to Scale Drawings | 比例尺介绍
A scale drawing is a representation of an object or area in which all dimensions have been shrunk or enlarged by the same ratio. The scale tells us the relationship between the lengths on the drawing and the real lengths. For instance, a scale of 1 : 100 means 1 cm on the drawing represents 100 cm (or 1 m) in reality. Common map scales include 1 : 25 000 or 1 : 50 000.
比例尺图是对物体或区域的表示，图中所有尺寸都按相同比例缩小或放大。比例尺告诉我们在图上长度与实际长度之间的关系。例如，比例尺 1 : 100 表示图上 1 cm 代表现实中的 100 cm（即 1 m）。常见的地图比例尺有 1 : 25 000 或 1 : 50 000。
| Scale | Meaning |
|---|---|
| 1 : 100 | 1 cm on map = 100 cm = 1 m in real life |
| 1 : 25 000 | 1 cm on map = 25 000 cm = 250 m in real life |
| 1 : 50 000 | 1 cm on map = 50 000 cm = 500 m in real life |
计算实际距离的公式很简单：图上长度乘以比例尺的分母，再转换为适当的单位。理解比例尺对于准确绘制和解读比例尺图至关重要。
8. Making a Scale Drawing | 制作比例尺图
To make a scale drawing, you need to know the real distances and the chosen scale. Convert all real distances to the same unit, then divide by the scale factor to find the drawing lengths. For example, a room is 4 m long and 3 m wide. Using a scale of 1 : 100, the drawing length is 4 m ÷ 100 = 0.04 m = 4 cm, and the width is 3 cm. Then you draw a 4 cm by 3 cm rectangle using a ruler and a protractor for right angles.
制作比例尺图时，你需要知道实际距离和选定的比例尺。将所有实际距离转换为相同单位，然后除以比例尺因子得到图上长度。例如，一个房间长 4 m、宽 3 m，使用比例尺 1 : 100，图上长度为 4 m ÷ 100 = 0.04 m = 4 cm，宽为 3 cm。然后用直尺和量角器画出一个 4 cm × 3 cm 的矩形（用直角保证形状正确）。
Drawing length = Actual length ÷ Scale factor
比例尺图必须保持角度一致，通常要使用量角器来测量方向，特别是在结合方位角绘制地图时。
9. Finding Distances from Scale Drawings | 从比例尺图中求距离
To find a real distance from a scale drawing, measure the length on the drawing with a ruler, then multiply by the scale factor. If a map uses a scale of 1 : 20 000 and the distance between two points on the map is 6.5 cm, the real distance is 6.5 cm × 20 000 = 130 000 cm = 1 300 m = 1.3 km. Always remember to convert the final answer to a sensible unit such as metres or kilometres.
要从比例尺图中求出实际距离，先用直尺测量图上的长度，再乘以比例尺因子。如果地图使用 1 : 20 000 的比例尺，图上两点之间的距离为 6.5 cm，则实际距离为 6.5 cm × 20 000 = 130 000 cm = 1 300 m = 1.3 km。请务必记住最终结果要转换为合适的单位，如米或千米。
Actual distance = Drawing length × Scale factor
确保测量精度，使用毫米刻度直尺读取最接近的毫米值，再按比例计算。该技能在解读建筑平面图、地图以及技术图纸时非常实用。
10. Scale Drawings with Bearings | 方位角结合比例尺的应用
In navigation and surveying, bearings and scale drawings are used together. You might be given distances and bearings for a journey and asked to construct a map. For example, a ship sails 5 km on a bearing of 060°, then 3 km on a bearing of 150°. By choosing a scale such as 1 cm : 1 km and using a protractor, you can draw both legs and find the final position. You can then measure the return bearing and distance directly from the drawing.
在导航和测量中，方位角和比例尺常一起使用。你可能会得到一段旅程中各段航程的距离与方位角，并据此绘制地图。例如，一艘船沿 060° 方位角航行 5 km，再沿 150° 方位角航行 3 km。选择比例尺（如 1 cm : 1 km），并使用量角器，就能画出这两段航程，并确定最终位置。然后可以直接从图上测量返回的方位角和距离。
This combined approach lets you solve complex navigation problems without using trigonometry, simply by drawing accurately and measuring. It is a core skill in the KS3 Cambridge mathematics course and provides a strong foundation for later topics like vectors and trigonometry.
这种结合绘图的方法让你无需使用三角函数就能解决复杂的导航问题，只需精确作图并测量即可。这是剑桥初中数学 KS3 阶段的核心技能，也为后续学习向量和三角学打下坚实基础。
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