📚 Bearings and Scale Drawings | 方位角与比例绘图
Bearings are a way of describing direction. They are widely used in navigation, map reading and surveying. A bearing is always measured clockwise from north, written as a three‑digit angle. In this article, you will learn how to find, draw and use bearings, and how to combine them with scale drawings to solve real‑world problems.
方位角是用来描述方向的一种方法，广泛应用于航海、地图阅读和测量中。方位角总是从正北方向开始顺时针测量，并用三位数字表示。本文将教你如何找出、绘制和使用方位角，以及如何将其与比例绘图结合，解决实际问题。
1. What is a Bearing? | 什么是方位角？
A bearing tells you the direction of one point from another. Instead of saying 'north‑east' or 'south‑west', we use a precise angle measured in degrees. Bearings are always given as three figures, for example 045° (north‑east) or 230° (south‑west). The starting direction is always true north.
方位角表示从一点到另一点的方向。我们不再使用“东北”或“西南”这样模糊的说法，而是用一个精确的度数来表示。方位角始终用三位数字表示，例如 045°（东北）或 230°（西南）。起始方向总是正北。
On diagrams, north is usually shown by an arrow pointing straight up. When we measure a bearing, we place the centre of the protractor at the starting point and line up 0° with the north line. Then we read the angle clockwise until we reach the line joining the two points.
在示意图中，北方通常用指向上方的箭头表示。测量方位角时，我们将量角器的中心放在起点，让 0° 对准北线，然后按顺时针方向读出到达两点连线处的角度。
2. The Three‑Figure Bearing Rule | 三位数方位角规则
Bearings must always be written with three digits. If the angle is less than 100°, we add a zero in front. For example, a bearing of 72° is written as 072°. A bearing of 5° becomes 005°. This rule avoids confusion and makes bearings easy to communicate.
方位角必须始终用三位数字书写。如果角度小于 100°，就在前面加上零。例如，72° 的方位角写成 072°，5° 写成 005°。这条规则能避免混淆，也便于交流。
There are two more important rules: a bearing is always measured from north, and always measured in a clockwise direction. You should never measure anticlockwise, even if the angle would be smaller. This is why a bearing can range from 000° up to 360°.
还有两条重要规则：方位角总是从正北开始，总是按顺时针方向测量。绝不能逆时针测量，即使那样角度会更小。这就是为什么方位角的范围可以从 000° 到 360°。
3. Measuring a Given Bearing | 测量已知方位角
To measure the bearing of point B from point A, draw a north line at A. Place your protractor over A with 0° pointing north. Follow the clockwise scale until the line AB meets the protractor. Read the angle; this is the bearing of B from A. Always write it with three digits.
要测量从点 A 看点 B 的方位角，首先在点 A 处画一条北线。把量角器放在 A 上，让 0° 对准北方，沿着顺时针刻度找到直线 AB 与量角器的交叉点，读出角度。这就是从 A 到 B 的方位角，一定要写成三位数。
If the line AB is exactly north, the bearing is 000°. If it is east, the bearing is 090°. South gives 180° and west gives 270°. These four are sometimes called the cardinal bearings.
如果直线 AB 恰好指向正北，方位角是 000°；正东是 090°；正南是 180°；正西是 270°。这四个方向有时被称为基本方位角。
4. Drawing a Bearing | 绘制方位角
When you need to draw a point from a known position, start by marking the starting point and drawing a clear north line. Use a protractor to measure the required angle clockwise from north. Draw a light construction line along this direction. Then use a ruler to mark the correct distance, often using a scale drawing.
当你需要从一个已知位置绘制另一个点时，先标出起点并画一条清晰的北线。用量角器从北方开始顺时针量出所需的角度，沿此方向画一条轻细的辅助线，然后使用直尺标出正确距离（通常会用到比例绘图）。
Always check that your protractor is aligned correctly: the centre must be exactly on the starting point and the 0° line must be exactly on north. Even a small shift can cause large errors over long distances.
一定要检查量角器是否对准：中心必须精确放在起点上，0° 线必须精确对准北方。即使微小的偏移，在远距离下也会导致很大的误差。
5. Back Bearings | 反向方位角
A back bearing tells you the direction from the final point back to the starting point. To find a back bearing, add 180° to the original bearing if it is less than 180°, or subtract 180° if it is 180° or more. For example, the back bearing of 035° is 215°; the back bearing of 200° is 020°.
反向方位角告诉你从终点回到起点的方向。求反向方位角的方法：如果原方位角小于 180°，就加上 180°；如果大于或等于 180°，就减去 180°。例如，035° 的反向方位角是 215°；200° 的反向方位角是 020°。
You can check a back bearing with a diagram. Draw both north lines at the two points (they are parallel). The angle measured at B back to A should be the supplementary angle, which gives the 180° rule.
你可以通过作图来检查反向方位角。在两个点处分别画出北线（它们是平行的），在点 B 处测量回到点 A 的角度，得到的正是互补角，由此验证了 180° 规则。
6. Scale Drawings – Linking Distance and Direction | 比例绘图——连接距离与方向
A scale drawing shows real objects or distances reduced or enlarged by a fixed ratio. For example, a map with scale 1 cm : 10 km means every 1 cm on the map represents 10 km in real life. Bearings and scale drawings are often used together to plot journeys or find unknown positions.
比例绘图将实物或真实距离按固定比例缩小或放大。例如，比例尺为 1 cm : 10 km 的地图，意味着图上每 1 cm 代表实际的 10 km。方位角和比例绘图常常一起使用，用来绘制行程或寻找未知位置。
The scale can be given as a ratio (1:50000), a statement (1 cm represents 500 m), or a scale bar. You must be able to convert between measurements on the drawing and real‑life distances using multiplication or division.
比例尺可以以比率（1:50000）、文字说明（1 cm 代表 500 m）或比例线条的形式给出。你必须能够通过乘法或除法，在图示尺寸与实际距离之间进行转换。
7. Converting Real Distances Using a Scale | 根据比例尺转换实际距离
To find the real‑ e scale is 1 cm : 5 m and the drawn length is 3.2 cm, the real distance is 3.2 × 5 = 16 m.
要计算比例图中的实际距离，先用尺子量出图纸上的长度，然后乘以比例因子。如果比例尺是 1 cm : 5 m，图上长度是 3.2 cm，那么实际距离就是 3.2 × 5 = 16 m。
factor. A real length of 45 m with scale 1 cm : 3 m becomes 45 ÷ 3 = 15 cm on the drawing.
反过来，要把实际距离转化为图纸长度，就将实际距离除以比例因子。实际长度为 45 m，比例尺为 1 cm : 3 m，那么在图上就是 45 ÷ 3 = 15 cm。
Always state your scale clearly on your diagram. A common mistake is to forget units — make sure the units on both sides of the scale match (e.g. both in cm, or cm to km converted properly).
务必在图上清晰地标明比例尺。一个常见的错误是忘记单位——确保比例尺两边的单位一致（例如都用厘米，或者把 cm 与 km 的关系换算正确）。
8. Worked Example – Finding a Bearing | 范例——求方位角
Problem: A lighthouse L is 8 km east and 15 km north of a harbour H. Find the bearing of L from H.
问题：灯塔 L 在港口 H 以东 8 km、以北 15 km 处。求从 H 到 L 的方位角。
First, sketch a right‑ ) ≈ 28°. Since this angle is measured from north towards east, the bearing is simply 028°.
首先画一个直角三角形，H 为原点。北线垂直向上，向东的距离（8 km）水平向右。可以利用三角学求出 H 点处北线与 HL 连线的夹角：tan θ = 对边/邻边 = 8/15，因此 θ = tan⁻¹(8/15) ≈ 28°。因为这个角是从北方往东量的，所以方位角就是 028°。
Always round your final bearing to three digits and, if the angle is not exact, to the nearest degree unless told otherwise. In navigation, bearings are usually given as whole numbers.
最后要把方位角写成三位数，如果角度不是整数，通常四舍五入到最近的度数，除非另有说明。在航海中，方位角通常以整数表示。
9. Worked Example – A Scale Drawing Journey | 范例——比例绘图行程
Problem: A ship sails from port P on a bearing of 060° for 120 km to point Q, then on a bearing of 150° for 90 km to point R. Using a scale of 1 cm : 30 km, draw the journey and find the bearing and distance of R from P.
问题：一艘船从港口 P 出发，沿方位角 060° 航行 120 km 到达点 Q，然后沿方位角 150° 航行 90 km 到达点 R。使用比例尺 1 cm : 30 km 画出这次旅程，并求从 P 到 R 的方位角和距离。
measure 060° clockwise, draw a 4 cm line to Q. At Q, draw a north line parallel to the first, measure 150° clockwise, draw a 3 cm line to R. Join P to R.
首先，将距离转化为图纸长度：120 km ÷ 30 km/cm = 4 cm；90 km ÷ 30 km/cm = 3 cm。在 P 点画北线，顺时针量取 060°，画一条 4 cm 的线段到 Q。在 Q 点画一条与第一条平行的北线，顺时针量取 150°，画 3 cm 线段到 R。最后连接 P 和 R。
Measure the length PR on the drawing — from P, place a protractor at P with 0° on north; measure clockwise to PR. Suppose you get 098°. Then the bearing is 098° and the direct distance is 171 km.
测量图上线段 PR 的长度——假设为 5.7 cm，实际距离 = 5.7 × 30 = 171 km。要找到从 P 到 R 的方位角，将量角器放在 P 点，让 0° 对准北方，顺时针测量到 PR 直线，假设得到 098°，那么方位角就是 098°，直线距离为 171 km。
10. Common Mistakes and How to Avoid Them | 常见错误与避免方法
- Forgetting the three‑figure rule. Always write bearings as three digits. Write 007°, not 7°.
- 忘记三位数规则。 始终将方位角写成三位数，比如写成 007° 而不是 7°。
- Measuring anticlockwise. Bearings are always clockwise from north. If your protractor has an anticlockwise scale, ignore it and use the clockwise side.
- 逆时针测量。 方位角总是从正北开始顺时针测量。如果你的量角器有逆时针刻度，不要使用，只读顺时针那一侧。
- Misaligning the protractor. The north line must go through 0° and the centre must sit exactly on the point. Double‑check before drawing.
- 量角器未对准。 北线必须穿过 0° 刻度，量角器中心必须精确对准测点。在画线之前要再检查一遍。
- Confusing scale conversion. tead of multiplying (or vice versa). Write down the scale and the conversion step by step: '1 cm represents 5 m, so 1 cm = 5 m'.
- 比例换算错误。 将乘除搞反。在纸上写下比例和换算步骤：“1 cm 代表 5 m，所以 1 cm = 5 m”。
- Not drawing clear north lines. Each point must have its own north line, parallel to the first one. Use a ruler and set square to keep them parallel.
- 没有画出清晰的北线。 每个点都必须有各自的北线，并与第一条北线平行。使用直尺和三角板来确保平行。
11. Practice Questions – Test Yourself | 练习题——自我检测
Question 1: A plane flies from airport A for 200 km on a bearing of 140° to point B. Find the bearing of A from B.
问题 1：一架飞机从机场 A 出发，200 km 沿方位角 140° 飞到点 B。求从 B 回到 A 的方位角。
nswer: Bearing of A from B = 140° + 180° = 320°. (Since 140° is less than 180°, add 180°.)
答案：从 B 到 A 的方位角 = 140° + 180° = 320°。（因为 140° < 180°，所以加 180°。）
Question 2: A map has a scale of 1 cm : 2.5 km. Two towns are 7.4 cm apart on the map. What is the real distance in km?
问题 2：一张地图的比例尺是 1 cm : 2.5 km。图上两镇相距 7.4 cm，实际距离是多少千米？
Answer: Real distance = 7.4 × 2.5 = 18.5 km.
答案：实际距离 = 7.4 × 2.5 = 18.5 km。
Question 3: A ship sails 50 km south, then 120 km east. Find the bearing of the final position from the start.
问题 3：一艘船向南航行 50 km，然后向东航行 120 km。求从起点到终点的方位角。
First, draw the right‑ s 180°, then add the 67° towards east, giving 180° + 67° = 247°. Thus the bearing is 247°.
首先画出直角三角形，南向 50 km，东向 120 km。从南方向东的夹角满足 tan θ = 120/50 = 2.4，θ ≈ 67°。方位角是从北开始算的：正南是 180°，再向东转 67°，得到 180° + 67° = 247°。所以方位角是 247°。
12. Summary and Key Points | 总结与要点
- Bearings are measured clockwise from north and written with three digits.
- 方位角从正北开始顺时针测量，并用三位数字书写。
- A back bearing is found by adding or subtracting 180°.
- 反向方位角可通过加或减 180° 得到。
- Scale drawings use a ratio to represent real distances; always multiply or divide correctly.
- 比例绘图用比率表示真实距离；要正确进行乘除运算。
- Protractor and ruler skills are essential — check alignment and north lines every time.
- 熟练使用量角器和直尺至关重要——每次都需检查对位情况和北线。
- Practice with right‑angled triangles and trigonometry to find bearings when given two perpendicular legs.
- 当已知两个垂直方向的边长时，通过直角三角形和三角学练习求方位角。
Published by TutorHao | Cambridge KS3 Mathematics Revision Series | aleveler.com
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