📚 Bearings, Scale Drawings and Maps | 方位角、比例图和地图
Bearings, scale drawings and maps are essential tools in geometry that allow us to describe directions precisely and represent real-world distances on paper. This topic consistently appears in IGCSE Edexcel exams, testing your ability to measure, calculate and interpret these representations accurately. In this revision guide, we will break down the key concepts step by step, from three‑figure bearings to constructing scale diagrams, ensuring you are fully prepared for any question that comes your way.
方位角、比例图和地图是几何学中的重要工具，使我们能够精确描述方向，并将真实世界的距离呈现在纸上。这个主题经常出现在 IGCSE Edexcel 考试中，考察你准确测量、计算和解读这些表示方法的能力。在本复习指南中，我们将逐步分解关键概念，从三位数方位角到绘制比例图，确保你为任何可能的题目做好充分准备。
1. What Are Bearings? | 什么是方位角？
A bearing is a way of describing the direction of one point from another, measured in a clockwise direction from the North line. Bearings are always given as three‑figure numbers (e.g. 045° instead of 45°) and are measured in degrees. They are used extensively in navigation and map reading.
方位角是一种描述从一个点到另一个点方向的方法，从北线开始按顺时针方向测量。方位角始终表示为三位数（例如 045° 而非 45°），并以度为单位。它们广泛用于导航和地图判读。
The bearing of B from A means we stand at A, face North, then turn clockwise until we are facing B. The angle turned is the bearing. Bearings must always be expressed with three digits, using leading zeros when necessary.
从 A 到 B 的方位角意味着我们站在 A 处，面向北，然后顺时针转动，直到面对 B。转过的角度就是方位角。方位角必须始终用三位数字表示，必要时在前面补零。
2. Measuring and Drawing Bearings | 测量与绘制方位角
To measure a bearing on a diagram, first draw a North line through the starting point. Place your protractor with its centre on the point and 0° pointing North. Measure the angle clockwise from North to the line joining the two points. If the angle is less than 100°, remember to add the leading zeros, e.g. write 073°.
要在图上测量方位角，首先通过起点画一条北线。将量角器的中心放在该点上，0° 指向北。从北开始顺时针方向测量与连接两点的线段之间的夹角。如果角度小于 100°，记得在前面补零，例如写作 073°。
When drawing a bearing from a point, start with a clear North line. Use a ruler and protractor to mark the required clockwise angle, then draw a straight line through the point. Label the angle clearly and ensure it is given as three figures.
从一点绘制方位角时，首先要有一条清晰的北线。使用直尺和量角器标记所需的顺时针角度，然后通过该点画一条直线。清晰地标注角度，并确保其为三位数。
3. Back Bearings | 后方位角
The back bearing is the direction you would take to return to your starting point. If you know the bearing of B from A, the back bearing of A from B is found by adding or subtracting 180°. If the original bearing is less than 180°, add 180°; if it is more than 180°, subtract 180°. The result must still be a three‑figure bearing.
后方位角是你返回起点的方向。如果你知道从 A 到 B 的方位角，那么从 B 到 A 的后方位角就是原方位角加上或减去 180°。如果原方位角小于 180°，则加 180°；如果大于 180°，则减 180°。结果仍须是三位数的方位角。
aring from the lighthouse back to the boat is 120° + 180° = 300°. Always check the diagram—back bearings are often used in ships and aircraft navigation problems.
例如，如果灯塔从一船只的方位角是 120°，那么从灯塔返回船只的后方位角就是 120° + 180° = 300°。始终对照图示检查——后方位角常用于船舶和飞机的导航问题。
4. Scale Drawings: Understanding the Scale | 比例图：理解比例尺
A scale drawing represents an object or area with dimensions proportionally smaller or larger than in real life. The scale is typically given as a ratio, such as 1 : 500, meaning that 1 cm on the drawing represents 500 cm (or 5 m) in reality. Scales can also be given as '1 cm represents 2 km' or similar statements.
比例图是将物体或区域的尺寸按比例缩小或放大后的表示。比例尺通常以比率形式给出，例如 1 : 500，表示图上的 1 cm 代表实际中的 500 cm（即 5 m）。比例尺也可能以“1 cm 代表 2 km”之类的表述给出。
When working with scales, always convert all measurements to the same units before calculating. For example, if a map uses a scale of 1 : 25 000, then 1 cm on the map equals 25 000 cm in reality, which is 250 m. Setting up a proportion will help you find unknown distances.
使用比例尺时，在计算之前务必将所有测量值转换为相同单位。例如，如果地图的比例尺是 1 : 25 000，那么图上的 1 cm 等于实际中的 25 000 cm，即 250 m。建立比例式有助于求出未知距离。
5. Converting Lengths and Areas Using Scales | 使用比例尺转换长度和面积
For a linear scale factor k (e.g. 1 : k), a length on the drawing is multiplied by k to obtain the real‑life length. If the scale is given as 1 cm to m km, first convert m km to cm, then use the equivalent ratio.
对于线性比例因子 k（例如 1 : k），将图上的长度乘以 k 即可得到实际长度。如果比例尺是 1 cm 表示 m km，请先将 m km 转换为 cm，再使用等效比率。
For areas, the scale factor is k (200)² = 3 × 40 000 = 120 000 cm². Convert this to m² by dividing by 10 000, giving 12 m².
对于面积，比例因子是 k²。如果平面图上一个花园的面积为 3 cm²，比例尺为 1 : 200，实际面积就是 3 × (200)² = 3 × 40 000 = 120 000 cm²。除以 10 000 转换为 m²，得到 12 m²。
6. Interpreting Maps with Scale and Bearings | 利用比例尺和方位角解读地图
Exam questions often combine bearings and scale on a map. You may be asked to measure the distance between two points on a map, then use the scale to find the real distance, or to determine the bearing of one point from another.
考试题目经常将方位角和比例尺结合在一张地图上。你可能会被要求测量地图上两点之间的距离，然后利用比例尺求出实际距离，或者确定一点相对于另一点的方位角。
Always use a sharp pencil and a ruler for measurement. When measuring a bearing on a map, first draw the North line accurately; it is often given by the map grid or a separate arrow. For distance, measure in centimetres, then multiply by the scale factor.
测量时务必使用削尖的铅笔和直尺。在地图上测量方位角时，首先要准确画出北线；北线通常由地图网格或单独的箭头表示。对于距离，以厘米为单位测量，然后乘以比例因子。
7. Constructing an Accurate Scale Drawing | 绘制精确的比例图
In many IGCSE problems, you need to create a scale diagram from given instructions. You will be provided a starting point and a series of bearings and distances. Follow each leg carefully: draw the North line, measure the bearing, draw a line along that direction, use the scale to mark the correct length, and label the new point.
在许多 IGCSE 问题中，你需要根据给定的指示创建比例图。题目会给出一个起点以及一系列方位角和距离。依次仔细完成每个航段：画北线，测量方位角，沿该方向画线，利用比例尺标出正确长度，然后标记新点。
After drawing the journey, you may need to find the bearing and distance of the final point from the starting point. This is done by joining the start and finish points with a straight line, then measuring the angle clockwise from North and the length of that line. Convert the measured length back to a real distance using the scale.
画出整个行程后，你可能需要找出终点相对于起点的方位角和距离。方法是用一条直线连接起点和终点，然后顺时针测量该线与北方向之间的角度以及该线的长度。再利用比例尺将测量长度换算回实际距离。
8. Using Trigonometry in Scale and Bearing Problems | 在比例与方位角问题中使用三角学
When a diagram forms a right‑angled triangle, you can apply Pythagoras' theorem or trigonometric ratios (sine, cosine, tangent) to find missing lengths or angles instead of relying solely on measurement. This is especially useful when a scale drawing is not required or when an accurate mathematical answer is needed.
当图形构成直角三角形时，你可以应用勾股定理或三角比（正弦、余弦、正切）来求缺失的边长或角度，而不完全依赖测量。这在不需要比例图或需要精确数学答案时尤为有用。
A typical problem: a ship sails 8 km on a bearing of 110°, then 5 km on a bearing of 200°. Find its distance and bearing from the start. Draw a sketch and break the journey into right‑angled components. Use sine and cosine to find the horizontal and vertical displacements, then find the resultant vector.
一个典型问题：一艘船沿 110° 方位角航行 8 km，然后沿 200° 方位角航行 5 km。求它离起点的距离和方位角。画出草图，将行程分解为直角分量。利用正弦和余弦求出水平和垂直位移，再求出合向量。
9. Common Mistakes and How to Avoid Them | 常见错误及其避免方法
One frequent error is forgetting to give bearings as three‑figure numbers. Writing 60° instead of 060° will lose a mark. Another mistake is measuring the angle anticlockwise from North; always remember bearings are clockwise.
一个常见错误是忘记将方位角表示为三位数。写 60° 而不是 060° 将会失分。另一错误是从北开始逆时针测量角度；始终记住方位角是顺时针的。
Scale errors often happen when units are mixed up. Always convert all distances to the same unit before applying the scale. When using a map, check whether the scale is given as '1 cm represents …' or as a ratio like 1 : 50 000, and work consistently. Finally, be careful with back bearings: adding or subtracting 180° incorrectly can lead to an answer well outside the 000°–360° range.
当单位混淆时常发生比例尺错误。在应用比例尺之前，务必将所有距离转换为相同的单位。使用地图时，检查比例尺是以“1 cm 代表 …”的形式给出，还是像 1 : 50 000 这样的比率，并始终一致地进行计算。最后，小心后方位角：错误地加减 180° 可能导致答案远远超出 000°–360° 的范围。
10. Exam Tips for Bearings, Scale Drawings and Maps | 方位角、比例图和地图的应试技巧
Always bring a good quality protractor, a sharp pencil, a ruler and a pair of compasses to the exam. For scale drawing questions, work on the diagram lightly in pencil first, then ink in your final lines once you are sure. Show all your steps clearly, including the conversion of distances using the scale.
考试时务必携带一把质量好的量角器、一支削尖的铅笔、一把直尺和一个圆规。对于比例图问题，先用铅笔在图上轻轻作业，确定后再用墨水笔描出最终线条。清楚地展示所有步骤，包括使用比例尺进行的距离换算。
When a question asks for the bearing of A from B, read the phrasing carefully—it is easy to measure the wrong bearing. Draw your own large, clear North lines at each point, and mark the angles legibly. If the protractor scale is confusing, double‑check whether you are reading clockwise or anticlockwise. Finally, always write your final bearing with three digits.
当题目要求求从 B 到 A 的方位角时，仔细阅读措辞——很容易测错方位角。在每个点画出自己大而清晰的北线，并清晰地标注角度。如果量角器的刻度令人困惑，请仔细检查你是在顺时针还是逆时针读取。最后，始终用三位数书写最终方位角。
| Scale expression | Meaning | 1 cm on paper equals |
|---|---|---|
| 1 : 100 | 1 cm represents 100 cm | 1 m |
| 1 : 10 000 | 1 cm represents 10 000 cm | 100 m |
| 1 : 250 000 | 1 cm represents 2.5 km | 2.5 km |
| 1 cm to 5 km | 1 cm represents 5 km | 5 km |
Bearings and scale drawings are highly visual and practical topics. Mastering them not only boosts your exam confidence but also builds a foundation for more advanced geometry, such as vectors and real‑world navigation. Regular practice with protractors and rulers is key.
方位角和比例图是高度可视且实用的主题。掌握它们不仅能提升你的考试信心，也为更高级的几何学（例如向量和现实世界导航）奠定基础。经常使用量角器和直尺进行练习是关键。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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