📚 Bearings and Scale Drawings | 方位角与比例绘图问题求解
Bearings and scale drawings are two connected skills in the Cambridge KS3 mathematics curriculum. Bearings give a precise direction from one point to another using angles measured clockwise from north. Scale drawings allow us to represent large real-world distances on a small piece of paper using a fixed ratio. In this article, you will learn how to read, measure and draw three-figure bearings, how to use map scales correctly, and how to combine both skills to solve navigation and construction problems. These ideas appear regularly in Cambridge Checkpoint tests and progression papers.
方位角和比例绘图是剑桥 KS3 数学课程中两项相互关联的技能。方位角用从正北方向顺时针测量的角度表示从一个点到另一个点的精确方向。比例绘图则使用固定比例，把巨大的实际距离缩小到一张纸上。本文将学习如何读取、测量和绘制三位数方位角，如何正确使用地图比例尺，以及如何综合运用这两种技能解决导航和作图问题。这些内容经常出现在剑桥 Checkpoint 测试和进阶试卷中。
1. What Is a Bearing? | 什么是方位角
A bearing is an angle measured in degrees that tells you the direction from one point to another. It is always measured clockwise from the north direction. In navigation, surveying and map reading, bearings are written as three-digit numbers, such as 045°, 120° or 310°. This rule removes confusion when directions are read quickly.
方位角是以度为单位的角度，用于表示从一个点到另一个点的方向。它总是从正北方向起，按顺时针方向测量。在导航、测量和地图阅读中，方位角写成三位数，例如 045°、120° 或 310°。这一规则避免了快速读取方向时产生混淆。
There are four common reference bearings: north is 000°, east is 090°, south is 180° and west is 270°. If a bearing is due northeast, it is written as 045° rather than 45°. This three-figure format is required in Cambridge answers.
有四个常见的参考方位角：正北是 000°，正东是 090°，正南是 180°，正西是 270°。如果方向是东北，则写成 045° 而不是 45°。剑桥考试的答案要求使用这种三位数格式。
2. Three-Figure Bearings | 三位数方位角
A three-figure bearing means the angle must have exactly three digits. If the angle is less than 100°, you add a zero in front. For example, 5° becomes 005°, 47° becomes 047° and 83° becomes 083°. If the angle is 100° or more, it already has three digits.
三位数方位角是指角度必须恰好有三个数字。如果角度小于 100°，就在前面加一个零。例如 5° 写作 005°，47° 写作 047°，83° 写作 083°。如果角度为 100° 或更大，则已经是三位数。
Bearings can never be greater than 360°. If you measure an angle of 365° by mistake, it means the correct bearing is 005° because you have gone once around the circle plus an extra 5°. Always check that your final bearing lies between 000° and 360°.
方位角永远不会大于 360°。如果你错误地测得 365°，说明正确方位角是 005°，因为你绕圆一周后又多走了 5°。始终检查最终方位角是否介于 000° 和 360° 之间。
3. Measuring Bearings with a Protractor | 使用量角器测量方位角
To measure a bearing from a point A to a point B, first draw a vertical north line through point A. Place the centre of the protractor at A, lining up 000° with the north line. Then measure clockwise until the line AB is reached. Read the angle on the outer scale if the protractor is placed with north pointing up.
要测量从点 A 到点 B 的方位角，首先通过点 A 画一条竖直的北方向线。将量角器的中心放在 A 上，使 000° 与北方向线对齐。然后顺时针测量，直到到达线段 AB。如果量角器的北方向朝上，请读取外圈刻度。
A common mistake is to measure anticlockwise or to use the wrong scale on a semicircular protractor. Always ask yourself: does my angle turn clockwise from north? If not, you may need to subtract your measurement from 360°.
一个常见错误是逆时针测量，或者在半圆量角器上读错了刻度。始终问自己：我的角度是从正北顺时针旋转的吗？如果不是，可能需要用 360° 减去你的测量值。
4. Back Bearings | 反向方位角
A back bearing is the bearing from point B back to point A. It is the opposite direction of the original bearing. To find a back bearing, add 180° if the original bearing is less than 180°, or subtract 180° if it is 180° or more. For example, the back bearing of 060° is 240°, and the back bearing of 250° is 070°.
反向方位角是从点 B 返回点 A 的方位角。它是原方位角的相反方向。求反向方位角的方法是：若原方位角小于 180°，则加上 180°；若大于或等于 180°，则减去 180°。例如 060° 的反向方位角是 240°，250° 的反向方位角是 070°。
from B to A, then the forward bearing from A to B must be 130° because 310° − 180° = 130°.
反向方位角在地图作业中很有用，比如你从 A 走到 B，然后需要返回 A。如果从 B 到 A 的罗盘读数是 310°，那么从 A 到 B 的前方位角一定是 130°，因为 310° − 180° = 130°。
5. Scale Drawings: Representing Real Distances | 比例绘图：表示实际距离
resents 50 000 cm in real life. Since 50 000 cm = 500 m, we can say 1 cm represents 500 m.
比例绘图是一种平面图或地图，其中每条长度都乘以相同的比例因子。比例尺通常写成比的形式，例如 1 : 50 000。这表示图上的 1 cm 代表实际 50 000 cm。因为 50 000 cm = 500 m，所以可以说 1 cm 代表 500 m。
Some scales are given as statements, such as '1 cm represents 2 km'. You can convert this to a ratio by changing both units to the same measure: 1 cm represents 2 km means 1 cm represents 200 000 cm, so the ratio is 1 : 200 000.
有些比例尺以文字形式给出，例如 '1 cm 代表 2 km'。可以将其转换为比，将两个单位化为相同单位：1 cm 代表 2 km 意味着 1 cm 代表 200 000 cm，因此比为 1 : 200 000。
| Scale | 比例尺 | Meaning | 含义 |
|---|---|
| 1 : 10 000 | 1 cm represents 100 m | 1 cm 代表 100 m |
| 1 : 25 000 | 1 cm represents 250 m | 1 cm 代表 250 m |
| 1 : 50 000 | 1 cm represents 500 m | 1 cm 代表 500 m |
| 1 : 100 000 | 1 cm represents 1 km | 1 cm 代表 1 km |
6. Choosing and Using a Scale | 选择并使用比例尺
cm, a scale of 1 cm representing 50 m would work. This is because 400 m ÷ 50 m/cm = 8 cm.
选择比例尺时，首先要确定绘图应有多大，以及需要显示多少实际距离。例如，一块地长 400 m，而你想在一张最大长度为 8 cm 的纸上画出它，那么 1 cm 代表 50 m 的比例尺是合适的。因为 400 m ÷ 50 m/cm = 8 cm。
Once the scale is chosen, use it consistently for every length. Do not change the scale halfway through a question. If a diagram is already given, always check the stated scale before measuring distances. A line segment that appears 3 cm long may represent 150 m, 300 m or 1.5 km depending on the scale.
一旦选定了比例尺，就要对每条长度始终如一地使用它。不要在题目中途更换比例尺。如果题目已经给出图形，在测量距离之前一定要先查看标注的比例尺。一条看起来 3 cm 长的线段，根据比例尺不同，可能代表 150 m、300 m 或 1.5 km。
7. Drawing a Simple Scale Diagram | 绘制简单比例图
To draw a scale diagram, start by converting all real distances into drawing lengths using the scale. Then use a ruler and sharp pencil to draw the first line from the starting point at the correct bearing. For example, draw a vertical north line, then use a protractor to mark the required clockwise angle.
绘制比例图时，首先用比例尺把所有实际距离换算成图上长度。然后用直尺和削尖的铅笔，从起点沿正确方位角画出第一条线。例如，先画一条竖直的北方向线，再用量角器标出所需的顺时针角度。
From the end of the first line, draw the next leg of the journey using its own bearing and length. Do not rotate the page or redraw north in a different direction; each bearing is always measured from the north line at the current position. Make small, neat arcs or lines to show how the protractor was used.
从第一条线的终点出发，使用该段路程自己的方位角和长度绘制下一段。不要旋转页面，也不要在不同方向上重新画北方向线；每个方位角总是从当前位置的北方向线开始测量。可以画出小而整齐的弧线或标记线，显示量角器的使用方法。
8. Solving Problems with Bearings and Scale Drawings | 解决方位角与比例绘图问题
Many Cambridge KS3 questions ask you to find the final position of a ship, walker or aircraft after two or more movements. The best method is to draw each leg carefully on the same diagram using the given scale and bearings. After completing the drawing, measure the final distance and bearing from the starting point to the finish point.
许多剑桥 KS3 题目会要求找出轮船、步行者或飞机在两次或多次移动后的最终位置。最好的方法是在同一张图上，按照给定的比例尺和方位角仔细画出每一段路程。完成绘图后，测量从起点到终点的最终距离和方位角。
135°. The directions 045° and 135° make a 90° angle at B, so triangle ABC is right-angled.
例如，假设点 B 距离点 A 为 4 km，方位角为 045°；点 C 距离点 B 为 3 km，方位角为 135°。如果 1 cm 代表 1 km，则从 A 沿 045° 画 AB = 4 cm，再从 B 沿 135° 画 BC = 3 cm。045° 和 135° 在 B 处成 90° 角，因此三角形 ABC 是直角三角形。
AC = √4² + 3² = √25 = 5 km
After drawing the diagram, AC should measure exactly 5 cm. To find the bearing of C from A, measure the angle clockwise from north at A to the line AC. It should be approximately 081°. This combines Pythagoras' theorem with scale drawing and bearing work.
画图后，AC 应恰好为 5 cm。要求 C 相对于 A 的方位角，在 A 点测量从北方向顺时针到 AC 的角。该角度应约为 081°。这样就将勾股定理与比例绘图和方位角知识结合起来。
9. Worked Example: Navigation Problem | 例题：导航问题
A boat leaves harbour H on a bearing of 030° and travels 6 km to point P. It then changes direction and travels on a bearing of 120° for 8 km to point Q. Using a scale of 1 cm representing 2 km, draw the journey and find the bearing and distance from H to Q.
一艘船从港口 H 出发，沿 030° 方位航行 6 km 到达点 P。然后改变方向，沿 120° 方位航行 8 km 到达点 Q。使用 1 cm 代表 2 km 的比例尺，画出这次航行，并求从 H 到 Q 的方位角和距离。
H, then draw HP = 3 cm at 030°. At P, draw a new north line, then draw PQ = 4 cm at 120°.
首先换算距离：6 km 变为 6 ÷ 2 = 3 cm，8 km 变为 8 ÷ 2 = 4 cm。在 H 处画一条北方向线，然后沿 030° 画 HP = 3 cm。在 P 处画一条新的北方向线，再沿 120° 画 PQ = 4 cm。
a 3-4-5 right triangle. Therefore HQ = 5 cm, which represents 5 × 2 = 10 km in real life.
若从 P 处的北方向线测量，HP 与 PQ 的夹角为 120° − 030° = 90°。由于 HP = 3 cm，PQ = 4 cm，三角形 HPQ 是 3-4-5 直角三角形。因此 HQ = 5 cm，实际距离为 5 × 2 = 10 km。
To find the bearing of Q from H, measure the angle clockwise from the north line at H to HQ. On an accurate drawing this angle is approximately 083°. Always state the bearing as a three-figure value.
求 Q 相对于 H 的方位角时，在 H 处从北方向线顺时针测量到 HQ。在一张精确的图上，这个角度约为 083°。务必将方位角写成三位数。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One of the most common errors is forgetting that bearings are measured clockwise from north. If a student measures anticlockwise, the bearing will be too small or completely wrong. Always draw or imagine a small north arrow at the point where you are taking the measurement.
最常见的错误之一是忘记方位角是从北方向顺时针测量的。如果学生逆时针测量，方位角会偏小或完全错误。测量时，请始终在测量点画出或想象一个小北方向箭头。
100 m. Always write down the scale conversion before starting the question and keep the same units throughout.
另一个错误是比例尺使用不当。如果 1 cm 代表 250 m，那么 5 cm 代表 5 × 250 = 1250 m，而不是 5 × 100 m。开始做题前，务必将比例尺换算写下来，并在整个过程中保持相同单位。
Some learners forget to add or subtract 180° correctly for back bearings. A useful check is that the forward and back bearings should differ by exactly 180°. If they differ by 200° or 160°, there is an error in measurement or calculation.
有些学生在计算反向方位角时，忘记正确地加或减 180°。一个有用的检验方法是：前方方位角与反向方位角应相差恰好 180°。如果相差 200° 或 160°，则说明测量或计算有误。
Finally, do not measure the angle from a horizontal or vertical grid line unless the question says so. Bearing questions always use north as the zero line, not east or west. Read the question wording carefully to check which line is meant.
最后，除非题目另有说明，不要从水平或竖直网格线开始测量角度。方位角问题始终以正北为基准线，而不是正东或正西。仔细阅读题干，确认应该以哪条线为基准。
11. Exam Tips and Practice Questions | 考试技巧与练习题
In an exam, always bring a sharp pencil, a ruler, a protractor and a pair of compasses. Draw scale diagrams on plain or squared paper as instructed. Neat work helps you measure final answers accurately and earn method marks even if the final value is slightly off.
考试时务必携带削尖的铅笔、直尺、量角器和圆规。按要求在空白纸或方格纸上绘制比例图。清晰的作图有助于准确测量最终答案，即使最后数值略有偏差，也能获得方法分。
epresents 5 × 20 000 = 100 000 cm = 1 km. Check your answer by converting units carefully.
练习快速转换实际距离与图上长度。例如，比例尺为 1 : 20 000 时，地图上的 5 cm 代表 5 × 20 000 = 100 000 cm = 1 km。仔细换算单位，检验答案。
Try this quick practice: A drone flies from A to B, a distance of 12 km on a bearing of 070°. It then flies from B to C, a distance of 5 km on a bearing of 160°. Using a scale of 1 cm to 2 km, draw the route and find the bearing and distance from A to C. A careful construction should give about 13 km on a bearing of approximately 100°.
试做一道快速练习：一架无人机从 A 飞往 B，距离 12 km，方位角为 070°。随后它从 B 飞往 C，距离 5 km，方位角为 160°。使用 1 cm 代表 2 km 的比例尺，画出路线，并求从 A 到 C 的方位角和距离。仔细作图后，结果应约为 13 km，方位角约为 100°。
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