📚 Bearings and Scale Drawings | 方位角与比例绘图
This Cambridge KS3 Mathematics revision article develops the skills you need to interpret, measure and construct bearings, and to use scale drawings to model real journeys, positions and distances accurately.
这篇剑桥 KS3 数学复习文章将帮助你掌握解读、量测与绘制方位角的方法，并学会运用比例绘图来准确模拟真实行程、位置与距离。
Bearings appear in navigation, map reading, surveying and engineering, and they connect geometry with real-world problem solving. Because a bearing is a convention rather than a personal choice, every student must follow exactly the same rules: measure clockwise, start from north, and write three digits.
方位角出现在导航、地图阅读、测量与工程中，它把几何与真实问题解决联系起来。由于方位角是一种约定而非个人选择，每位学生都必须遵循完全相同的规则：从正北开始、顺时针量测、写成三位数。
1. What Are Bearings? | 什么是方位角
Bearings are used in navigation, map reading and surveying to describe direction. A bearing is always measured clockwise from the north direction, and it is always given as an angle between 000° and 360°.
方位角用于导航、地图阅读和测量中描述方向。方位角总是从正北方向顺时针量测，并且始终表示为 000° 到 360° 之间的角度。
For example, a ship sailing due east has a bearing of 090°, and a ship sailing due south has a bearing of 180°. A ship sailing due west has a bearing of 270°. Because the measurement always starts from north and turns clockwise, there is never any doubt about which way an angle is being read.
例如，一艘向正东行驶的船方位角为 090°，向正南行驶的船方位角为 180°，向正西行驶的船方位角为 270°。由于量测始终从正北开始并顺时针转动，因此在读数时永远不会产生方向歧义。
- North = 000° | 北方 = 000°
- East = 090° | 东方 = 090°
- South = 180° | 南方 = 180°
- West = 270° | 西方 = 270°
Two more conventions matter just as much. First, the bearing of a point is measured from a named starting point, so we say 'the bearing of B from A'. Second, if you reverse the direction of travel you do not get the same bearing, and this idea is developed in a later section.
还有两个约定同样重要。第一，某点的方位角必须从指定起点量测，所以我们说"从 A 看点 B 的方位角"。第二，如果方向反过来，得到的方位角并不相同，这一点将在后面的小节中展开。
2. Reading and Measuring Bearings | 读取和量测方位角
To measure the bearing of a point B from a point A, draw a clear north line at A. Place the protractor with its centre at A and 0° pointing north, then read the scale that increases clockwise.
若要量测从点 A 看点 B 的方位角，先在 A 点画出清晰的正北线。将量角器的中心放在 A 点，并让 0° 指向北方，然后读取顺时针增大的那一圈刻度。
Measure clockwise from this north line to the line AB. If the angle is less than 100°, write it as a three-digit number by adding a leading zero. If the angle is greater than 180°, you must be careful to read the reflex part of the circle rather than the smaller angle.
从这条北线顺时针量到 AB 线的角度。如果角度小于 100°，则在前方补零写成三位数。如果角度大于 180°，就必须注意读取圆周上的较大那一侧，而不是较小的角。
Bearing of B from A = 065°
从 A 看点 B 的方位角 = 065°
A useful check is to place the protractor on the north line and confirm that the zero mark, the centre of the protractor and the north arrow all line up. Slight misalignment is the most common reason for a measurement that is two or three degrees out.
一个实用的检查方法是：把量角器放在北线上，确认零刻度、量角器中心与指北箭头三者对齐。轻微的对不齐，是量测结果相差两三度的最常见原因。
3. Three-Digit Bearings Convention | 三位数方位角规则
Bearings are always written using three digits. This avoids confusion when reading directions aloud or writing them on charts, and it makes it obvious when an answer has not been completed properly.
方位角始终写成三位数字。这样可以避免在口头读数或在地图上书写时产生混淆，也能明显看出答案是否书写规范。
If a protractor gives an angle of 8°, the bearing is 008°. If the angle is 105°, the bearing is simply 105°. If the angle is 60°, the bearing is 060° and not 60°.
如果量角器读出的角度是 8°，方位角写作 008°。如果角度是 105°，方位角就直接写作 105°。如果角度是 60°，方位角写作 060°，而不是 60°。
Examiners regularly award marks for correct three-digit notation, so losing a mark for writing 45° instead of 045° is entirely avoidable. A bearing must also never be negative and never exceed 360°.
考官经常为正确的三位数写法给分，所以因为把 045° 写成 45° 而丢分是完全可避免的。方位角也绝不可以是负数，并且不能超过 360°。
| Compass direction | Bearing |
|---|---|
| North-east | 045° |
| South-east | 135° |
| South-west | 225° |
| North-west | 315° |
4. Compass Points and Common Bearings | 罗盘方位与常用方位角
The four cardinal points give the bearings 000°, 090°, 180° and 270°, while the four diagonal directions lie exactly halfway between them. Knowing these eight values by heart makes both drawing and checking much faster.
四个基本方位给出 000°、090°、180° 和 270°，而四个对角方向正好位于它们之间的一半位置。熟记这八个数值可以让绘图和检查都更快。
Between the cardinal points there are also intermediate directions such as north-north-east at 022.5° and east-north-east at 067.5°. You will not often need these in KS3, but recognising that they exist shows you understand that a bearing is a continuous measurement, not a label.
基本方位之间还有中间方向，例如北偏东北为 022.5°，东偏东北为 067.5°。在 KS3 阶段你通常不需要用到这些，但知道它们的存在说明你理解方位角是一种连续的量测，而不是一个名称标签。
the bearing is 180° minus the angle from south, so 30° east of south is 180° − 30° = 150°.
东北象限内的方位角就是从北转过的角度，因此"北偏东 40°"就是 040°。东南象限内的方位角等于 180° 减去从南偏过的角度，因此"南偏东 30°"就是 180° − 30° = 150°。
the bearing is 360° minus the angle from north, so 25° west of north is 360° − 25° = 335°.
西南象限内的方位角等于 180° 加上从南偏过的角度，因此"南偏西 40°"就是 180° + 40° = 220°。西北象限内的方位角等于 360° 减去从北偏过的角度，因此"北偏西 25°"就是 360° − 25° = 335°。
5. Bearings Between Two Points and Back Bearings | 两点间方位角与反方位角
The bearing of B from A and the bearing of A from B are never equal unless both lie on exactly the same north-south or east-west line. In general the two differ by 180°, and this is called the back bearing.
从 A 看点 B 的方位角与从 B 看点 A 的方位角一般并不相等，除非两点恰好在同一条南北线或东西线上。通常两者相差 180°，这个角称为反方位角。
If bearing < 180°, back bearing = bearing + 180°
若方位角 < 180°，反方位角 = 方位角 + 180°
If bearing > 180°, back bearing = bearing − 180°
若方位角 > 180°，反方位角 = 方位角 − 180°
= 245°. If the bearing of C from D is 310°, the bearing of D from C is 310° − 180° = 130°.
例如，从 A 看点 B 的方位角为 065°，那么从 B 看点 A 的方位角就是 065° + 180° = 245°。若从 D 看点 C 的方位角为 310°，则从 C 看点 D 的方位角是 310° − 180° = 130°。
An exact result of 180° or 360° should be written as 180° or 000°, never as 360°. Remember that 000° and 360° describe the same direction, but the convention is to write the smaller three-digit value.
如果结果正好是 180° 或 360°，应写作 180° 或 000°，绝不写成 360°。记住 000° 与 360° 表示同一方向，但按约定应写成较小的那个三位数值。
6. Angles on Parallel North Lines | 平行北线上的角度
When you draw north lines at two different points on the same map, those two north lines are parallel, because north is the same direction everywhere on a flat drawing. This gives you a powerful geometry tool for checking bearings.
当你在同一张地图的两个不同点上画北线时，这两条北线互相平行，因为在平面图上各处正北方向相同。这为你提供了一个检验方位角的有力几何工具。
The line joining the two points acts as a transversal across the two parallel north lines. The alternate angles formed are equal, which is exactly why the difference between a bearing and its back bearing is 180°.
连接两点的直线就像穿过两条平行北线的截线。由此形成的交错角相等，这正是方位角与反方位角相差 180° 的原因。
Co-interior angles between the parallel north lines add up to 180°. You can therefore find an unknown bearing by working with the angles at the second point instead of measuring again with a protractor.
两条平行北线之间的同旁内角之和为 180°。因此你可以利用第二个点处的角度来求未知方位角，而不必再次用量角器量测。
In a typical exam question you are told the bearing of B from A and asked for the bearing of A from B. Drawing the north lines and marking the alternate angles gives a complete geometrical justification, which is often worth more marks than a bare numeric answer.
在典型的考题中，题目给出从 A 看点 B 的方位角，要求从 B 看点 A 的方位角。画出北线并标出交错角，可以给出完整的几何理由，这通常比只写一个数字答案得分更高。
7. Scale and Scale Drawings | 比例与比例绘图
A scale drawing is an accurate diagram in which every length on the paper is a fixed fraction of the real length. The scale is usually written as a ratio such as 1 : 50 000 or in the form 1 cm : 2 km.
比例图是一种精确的图形，图中纸上的每一条长度都是真实长度的一个固定比例。比例尺通常写成 1 : 50 000 这样的比，或写成 1 cm : 2 km 这样的形式。
000 cm in reality. Since 50 000 cm = 500 m, we can also say 1 cm : 500 m, or 2 cm : 1 km.
当比例尺为 1 : 50 000 时，图上 1 cm 代表实际 50 000 cm。由于 50 000 cm = 500 m，也可以说 1 cm : 500 m，或 2 cm : 1 km。
m = 1 km, therefore 100 000 cm = 1 km. A scale of 1 : 100 000 is the same as 1 cm : 1 km.
比例尺之间的换算是常见错误来源，所以一定要仔细换算单位：100 cm = 1 m，1000 m = 1 km，因此 100 000 cm = 1 km。比例尺 1 : 100 000 等同于 1 cm : 1 km。
| Scale | 1 cm on paper represents | 中文说明 |
|---|---|---|
| 1 : 25 000 | 250 m | 图上 1 cm 代表 250 m |
| 1 : 50 000 | 500 m | 图上 1 cm 代表 500 m |
| 1 : 100 000 | 1 km | 图上 1 cm 代表 1 km |
| 1 cm : 2 km | 2 km | 图上 1 cm 代表 2 km |
8. Constructing a Scale Drawing | 绘制比例图的步骤
To construct an accurate scale drawing, first choose a sensible scale so that the final diagram fits comfortably on the page. Then draw a north line at the starting point and mark it clearly with an arrow.
要绘制准确的比例图，先选择一个合适的比例尺，使最终图形能舒适地画在纸上。然后在起点画出北线，并用箭头清晰标出方向。
Next, place the protractor at the starting point with 0° on the north line and mark the required bearing with a small dot. Draw a straight line from the start through that dot and measure the scaled length along it with a ruler.
接着，把量角器放在起点上，让 0° 对准北线，并在所需方位角处点一个小点。从起点经过该点画一条直线，再用直尺沿这条线量出按比例换算后的长度。
Repeat the process at the new point, remembering to draw a fresh north line there. All north lines on the same drawing are parallel and point the same way, which is essential for the geometry to work correctly.
在新点上重复这一过程，并记住在该处重新画一条北线。同一张图中所有北线都互相平行且方向一致，这是几何关系能够成立的关键。
Label every point, every given distance and every given bearing. An accurate, fully labelled diagram is much easier for an examiner to follow, and it also helps you spot your own mistakes before you write your final answer.
标注每一个点、每一个已知距离和每一个已知方位角。准确且标注完整的图形不仅便于考官阅读，也能帮助你在写出最终答案之前发现自己的错误。
9. Using Scale Drawings to Solve Journey Problems | 用比例图解决行程问题
A classic journey problem gives two or more legs, each described by a distance and a bearing, and asks for the direct distance and bearing back to the starting point. A scale drawing solves it quickly and visibly.
经典的行程问题会给出两段或更多段行程，每段用距离和方位角描述，并要求求出返回起点的直线距离与方位角。比例图可以快速而直观地解决这类问题。
Suppose a boat leaves port P on a bearing of 060° and travels 5 km to Q. It then turns and travels 4 km on a bearing of 150° to R. Choose a scale of 1 cm : 1 km, draw both legs accurately, then measure RP and the bearing of P from R.
假设一艘船从港口 P 出发，沿方位角 060° 航行 5 km 到达 Q，然后转向沿方位角 150° 航行 4 km 到达 R。选择比例尺 1 cm : 1 km，准确画出两段行程，然后量出 RP 的长度以及从 R 看点 P 的方位角。
al distance is about 6.4 km. Measuring the angle clockwise from the north line at R gives a bearing of approximately 279°, so the boat should sail on a bearing of about 279° to return to P.
图上量得 RP ≈ 6.4 cm，因此实际距离约为 6.4 km。从 R 点的北线顺时针量角，得到方位角约为 279°，所以船应以约 279° 的方位角航行返回 P。
cos 150° ≈ −3.46 km north. So R is about 6.33 km east and 0.96 km south of P, giving RP ≈ √6.33² + 0.96² ≈ 6.4 km.
你可以用坐标法检验：Q 在 P 以东 5 sin 60° ≈ 4.33 km、以北 5 cos 60° = 2.5 km；R 再向东 4 sin 150° = 2 km、向北 4 cos 150° ≈ −3.46 km。因此 R 位于 P 以东约 6.33 km、以南约 0.96 km，于是 RP ≈ √6.33² + 0.96² ≈ 6.4 km。
Always state the scale you used and the measured values to a sensible degree of accuracy. If the question asks for an answer to the nearest kilometre or the nearest degree, round at the final step, not in the middle of the working.
务必写出你使用的比例尺，并把量测值保留到合理的精度。如果题目要求精确到最近的千米或最近的度，应在最后一步取整，而不是在计算过程中取整。
10. Worked Examples | 例题精讲
ind the bearing of H from L. Since 118° < 180°, the back bearing is 118° + 180° = 298°.
例 1：从港口 H 看灯塔 L 的方位角是 118°。求从 L 看 H 的方位角。由于 118° < 180°，反方位角为 118° + 180° = 298°。
r park, to the nearest degree. Here tan θ = 7 ÷ 3, so θ ≈ 66.8°, giving a bearing of 067°.
例 2：一名徒步者位于停车场以东 7 km、以北 3 km。求从停车场看徒步者的方位角（精确到度）。这里 tan θ = 7 ÷ 3，所以 θ ≈ 66.8°，方位角为 067°。
cale 1 cm : 5 km, two villages are 4.2 cm apart on the paper. The real distance is 4.2 × 5
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