📚 Bearings and Scale Drawings | 方位角与比例图
Bearings and scale drawings are used in navigation, map reading and construction. In this revision guide, you will learn how to measure and use three-figure bearings, draw accurate scale diagrams, and solve return-journey problems. These skills are directly tested in the Cambridge KS3 and Checkpoint exam.
方位角和比例图在航海、地图阅读和建筑绘图中非常实用。本复习指南将帮助你学习三位数方位角的测量与使用、绘制精确的比例图，以及解决返程问题。这些技能在剑桥 KS3 与 Checkpoint 考试中会直接考查。
1. What Are Bearings? | 什么是方位角？
A bearing is a direction measured from the North direction, clockwise, in degrees. It tells you the angle through which you must turn from North to face a target. Bearings are always written with three digits, such as 045°, not 45°.
方位角是从正北方向顺时针测量的角度，以度为单位。它表示从正北转向目标方向需要转过的角度。方位角总是写成三位数，例如 045°，而不是 45°。
The basic compass directions have these bearings: North is 000°, East is 090°, South is 180°, and West is 270°. A bearing can never be 360° or more, because 360° is the same as 000°.
基本罗盘方向的方位角为：北 000°，东 090°，南 180°，西 270°。方位角不能达到 360° 或更大，因为 360° 等同回到正北，记作 000°。
Bearing = clockwise angle from North
方位角 = 从正北方向顺时针旋转的角度。
2. The Three-Digit Rule | 三位数规则
A bearing is always given as three digits. If the angle is less than 100°, place zeros in front to make three digits. For example, an angle of 7° becomes 007°, an angle of 35° becomes 035°, and an angle of 112° is written 112°.
方位角总是写成三位数字。如果角度小于 100°，就在前面补零，使其成为三位数。例如，7° 写作 007°，35° 写作 035°，112° 写作 112°。
The three-digit rule removes confusion. A bearing of 5° is written 005°, so it cannot be mistaken for 050° or 500°. In a diagram, this helps examiners and navigators read the direction instantly.
三位数规则可以消除混淆。5° 写成 005°，就不会被误认为 050° 或 500°。在图上，这有助于阅卷人和导航员快速读懂方向。
3. Measuring a Bearing with a Protractor | 用量角器测量方位角
To measure a bearing from a point A to a point B, draw a faint North line at A. Place the centre of the protractor at A with 0° pointing North. Measure clockwise from the North line to the line AB. Write the angle as a three-figure bearing.
要测量从点 A 到点 B 的方位角，先在 A 点画一条淡淡的北方向线。将量角器的中心放在 A 点，使 0° 对准正北。从北方向线顺时针量到线段 AB，并将角度写成三位数方位角。
bearing. For example, if the anticlockwise angle is 40°, the bearing is 360° - 40° = 320°.
务必检查量角器方向是否正确。如果你测得的是逆时针方向的小角，就要用 360° 减去这个角，得到顺时针方位角。例如，逆时针角为 40°，方位角就是 360° - 40° = 320°。
4. Back Bearings | 反方位角
The back bearing is the bearing of the return path, from B back to A. If the original bearing is less than 180°, add 180°. If the original bearing is 180° or more, subtract 180°. This is because a reverse direction is exactly half a turn, or 180°, different.
反方位角是从 B 回到 A 的返程方位角。如果原来的方位角小于 180°，就加上 180°；如果原来的方位角等于或大于 180°，就减去 180°。这是因为反向正好相差半圈，即 180°。
o A is 075° + 180° = 255°. If the bearing is 240°, the back bearing is 240° - 180° = 060°.
例如，从 A 到 B 的方位角是 075°，那么从 B 到 A 的反方位角为 075° + 180° = 255°。如果方位角是 240°，反方位角就是 240° - 180° = 060°。
Back bearing = bearing + 180° if bearing < 180°; bearing - 180° if bearing ≥ 180°
反方位角 = 原方位角 + 180°（当原方位角 < 180°）；原方位角 - 180°（当原方位角 ≥ 180°）。
5. Scale Drawing Basics | 比例图基础
A scale drawing shows a real object or journey at a smaller size. A map scale such as 1 : 25000 means 1 cm on the map represents 25000 cm, or 250 m, in real life. Use the same units when converting.
比例图以较小的尺寸表示真实物体或路程。例如地图比例尺 1 : 25000 表示图上 1 cm 代表实际 25000 cm，也就是 250 m。换算时必须使用相同单位。
mple, on a 1 : 50000 map, a line of 4 cm represents 4 × 50000 = 200000 cm = 2000 m = 2 km.
要根据图上长度求实际距离，用图上距离乘以比例因子。例如，在 1 : 50000 的地图上，4 cm 的线段代表 4 × 50000 = 200000 cm = 2000 m = 2 km。
Real distance = Map distance × Scale factor
实际距离 = 图上距离 × 比例因子。
6. Converting Real Distances to Map Distances | 实际距离与图上距离换算
: 20000, a real distance of 1 km, which is 100000 cm, is 100000 ÷ 20000 = 5 cm on the map.
如果你知道实际距离和比例尺，就用实际距离除以比例因子，得到图上距离。例如，比例尺为 1 : 20000，实际距离 1 km，即 100000 cm，那么图上距离为 100000 ÷ 20000 = 5 cm。
Always convert both lengths to the same unit before scaling. A common error is to mix units, such as dividing metres by a scale in centimetres. Write all lengths in cm first.
在换算前，一定要把两个长度换算成相同单位。常见的错误是单位混用，例如用米除以以厘米为单位的比例尺。应先把所有长度都写成厘米。
Map distance = Real distance ÷ Scale factor
图上距离 = 实际距离 ÷ 比例因子。
7. Drawing a Two-Stage Journey | 绘制两段旅程
A typical Checkpoint question asks you to draw a journey with two legs. For example, a ship sails 8 km on a bearing of 075°, then 5 km on a bearing of 150°. Choose a sensible scale, such as 1 cm : 1 km. Draw a North line at each stage and measure the next bearing from that North line.
典型的 Checkpoint 题目会要求你画出一段两段式旅程。例如，一艘船以 075° 的方位角航行 8 km，然后以 150° 的方位角航行 5 km。选择一个合适的比例尺，例如 1 cm : 1 km。在每一段起点都画出北方向线，并从新的北方向线开始测量下一段方位角。
Start at point A. Draw the first line 8 cm long at 075° from North. Label the end B. At B, draw a new North line parallel to the first. Measure 150° clockwise from this new North line, and draw the second line 5 cm long. Label the end C.
从 A 点开始。从北方向顺时针量 075°，画一条 8 cm 长的线段，终点标为 B。在 B 点画一条新的北方向线，与原北方向线平行。从这条新北方向线顺时针量 150°，画出第二条 5 cm 长的线段，终点标为 C。
To find the final distance from start to finish, measure the straight line from A to C and convert using the scale. To find the final bearing from A to C, draw a North line at A and measure clockwise to AC.
要得到从起点到终点的最终距离，可以测量 A 到 C 的直线距离，再按比例尺换算。要得到从 A 到 C 的最终方位角，在 A 点画出北方向线，顺时针量到 AC。
8. Finding the Return Bearing and Distance | 求返程方位角和距离
After drawing the journey, you may need the return bearing from C back to A. Draw a North line at C and measure the clockwise angle from North to line CA. Alternatively, use the back-bearing rule if the bearing from A to C is known.
画完旅程后，你可能需要求从 C 回到 A 的返程方位角。在 C 点画一条北方向线，从正北顺时针量到线段 CA。如果已经知道从 A 到 C 的方位角，也可以使用反方位角规则。
°. The return distance is the same as AC, because distance is scalar and does not depend on direction.
例如，如果图中显示从 A 到 C 的方位角是 111°，那么从 C 到 A 的返程方位角为 111° + 180° = 291°。返程距离与 AC 相同，因为距离是标量，与方向无关。
Check your diagram: the return bearing should point roughly opposite to the outward direction. If the outward bearing is between 000° and 180°, the back bearing is between 180° and 360°.
检查你的图：返程方位角应大致指向去程的相反方向。如果去程方位角在 000° 到 180° 之间，反方位角就在 180° 到 360° 之间。
9. Scale Factors and Area | 比例因子与面积
0 m². On the plan, each length is divided by 100, so the paper area is 2 m × 1.5 m = 3 m².
当一个图形被缩放时，长度要乘以比例因子 k，面积要乘以 k²。例如，地图比例尺为 1 : 100，一块 200 m × 150 m 的矩形实际场地面积为 30000 m²。在图纸上，每个长度都除以 100，因此纸面面积为 2 m × 1.5 m = 3 m²。
If the scale factor is 2, lengths double, but the area becomes four times larger. If the scale factor is ½, the area becomes one quarter of the original. This is useful when comparing maps
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