📚 Bearings | 方位角
Welcome to this revision article on bearings, directly linked to page 180 of your Cambridge Lower Secondary Mathematics coursebook. Here you will learn what bearings are, how to measure and draw them accurately, and how to apply them in navigation and scale diagrams. Whether you are working on Exercise 18.1 or checking your understanding before a checkpoint test, this guide will take you through every key point step by step.
欢迎阅读这篇关于方位角的复习文章，内容直接对应于剑桥初中数学教材第180页的练习。你将学习什么是方位角、如何准确测量和绘制方位角，以及如何在导航和比例图中应用它们。无论你正在完成练习18.1还是在 checkpoint 测试前查漏补缺，这篇指南都将带你逐步掌握每一个关键点。
1. What are Bearings? | 什么是方位角？
A bearing is a way of describing the direction of one point from another. It is always measured as an angle from the north direction, moving clockwise. Bearings are used in navigation, mapping and when giving directions on land or at sea.
方位角是描述一个点相对于另一个点的方向的方式。它总是从正北方向开始，以顺时针方向测量角度。方位角用于导航、绘制地图以及在陆地或海上指示方向。
In mathematics, a bearing is a three‑figure angle, so it is always written with three digits. For example, a bearing of 55 degrees is written as 055°, and a bearing due east is 090°.
在数学中，方位角是一个三位数的角度，因此始终用三位数字书写。例如，55度的方位角写作055°，正东方向的方位角为090°。
2. Three-Figure Bearings | 三位数方位角
Always write bearings using three figures. If the angle is less than 100°, put a zero or two zeros in front. This avoids confusion with ordinary angles and makes communication clear in navigation.
始终用三位数字表示方位角。如果角度小于100°，就在前面加一个或两个零。这样可以避免与普通角度混淆，并使导航中的信息传递清晰无误。
- North → 000°
- Northeast → 045°
- East → 090°
- Southeast → 135°
- South → 180°
- Southwest → 225°
- West → 270°
- Northwest → 315°
北 → 000°，东北 → 045°，东 → 090°，东南 → 135°，南 → 180°，西南 → 225°，西 → 270°，西北 → 315°。
3. How to Measure Bearings | 如何测量方位角
To find the bearing of point B from point A, place your protractor with its centre at A and the 0° mark pointing north. Read the angle clockwise from the north line to the line joining A to B. This angle is the bearing of B from A.
要找到点B相对于点A的方位角，将量角器的中心放在点A上，并将0°刻度对准正北方向。然后顺时针读出从正北线到连接A与B的直线之间的角度。这个角度就是点B相对于点A的方位角。
If the angle is acute, you still measure clockwise to get a value between 0° and 360°. For example, when B is to the east of A, the bearing is 090°.
即使角度是锐角，你仍然需要顺时针测量，以得到介于0°到360°之间的值。例如，当B位于A的正东方向时，方位角为090°。
4. How to Draw Bearings | 如何绘制方位角
To draw a bearing, start by drawing a north line at your starting point. Use a protractor to measure the required angle clockwise from north and mark a point on that direction. Always label the north line and write the bearing clearly.
要绘制方位角，首先在起点处画出正北线。用量角器从正北顺时针量出所需的角度，并在这条方向上标记一个点。务必标出正北线并清楚地写上方位角。
If you are asked to draw a ship sailing on a bearing of 120° for 5 cm, draw the north arrow, measure 120° clockwise, draw a ray and mark a point 5 cm along it. This forms part of a scale drawing.
如果要求绘制一艘船沿120°方位角航行5厘米，先画出正北箭头，顺时针量出120°，画一条射线，并在其上5厘米处标记点。这就构成了比例图的一部分。
5. Bearings and the Compass | 方位角与罗盘方向
Bearings are closely related to compass directions. The table below shows how standard compass points translate into three‑figure bearings. Learning these helps you estimate bearings mentally before using a protractor.
方位角与罗盘方向密切相关。下表显示了标准罗盘点如何转换为三位数方位角。学习这些内容有助于你在使用量角器之前心算估算方位角。
| Compass Point | Bearing |
|---|---|
| N | 000° |
| NNE | 022.5° |
| NE | 045° |
| ENE | 067.5° |
| E | 090° |
| SE | 135° |
| S | 180° |
| SW | 225° |
| W | 270° |
| NW | 315° |
记住：方位角总是从北顺时针测量，因此罗盘方向可以直接转换为角度。
6. Back Bearings | 反向方位角
The back bearing is the bearing you would use to return from point B to point A. It is found by adding 180° to the original bearing if the result is less than 360°, or subtracting 180° if the original is 180° or more. This works because the reverse direction is exactly half a turn away.
反向方位角是从点B返回点A所采用的方位角。如果原方位角加上180°后小于360°，则加上180°；如果原方位角大于或等于180°，则减去180°。这是因为反方向正好相差半圈。
° = 255°. If the bearing of X from Y is 210°, then the back bearing is 210° - 180° = 030°.
例如，B相对于A的方位角是075°。那么反向方位角，即A相对于B的方位角，就是075° + 180° = 255°。如果Y相对于X的方位角是210°，则反向方位角为210° - 180° = 030°。
7. Bearings in Scale Drawings | 比例绘图中的方位角
Many problems on page 180 require you to represent a journey using bearing lines and distances on a scale drawing. A scale is chosen, such as 1 cm to represent 1 km or 1 cm to represent 10 m. You then draw the path step by step, starting each new leg with a fresh north line at the point you have reached.
第180页的许多问题要求你在比例图上用方位线及距离表示一段行程。需选择比例尺，例如1厘米代表1公里或1厘米代表10米。然后逐步绘制路径，每开始一段新航程时，都必须在到达的点上重新画出正北线。
Accuracy is key. Always use a sharp pencil, place your protractor carefully and measure distances with a ruler. The final position can then be used to measure the return bearing and distance from the finish back to the start.
准确性至关重要。务必使用削尖的铅笔，仔细放置量角器，并用直尺测量距离。最终的到达点可以用来测量从终点返回起点的方位角和距离。
8. Worked Example: Finding a Bearing | 例题：求方位角
Problem: Town P is 8 km east and 3 km north of town Q. Find the bearing of P from Q.
问题：P镇在Q镇以东8公里、以北3公里处。求P相对于Q的方位角。
Solution: Draw a right‑ ed clockwise from north, which is exactly θ, so the bearing is 069° (to the nearest degree).
解：画一个直角三角形，水平直角边8公里，竖直直角边3公里。偏离正北方向的角度θ由tan θ = 对边/邻边 = 8/3 求得。因此θ = tan⁻¹(8/3) ≈ 69.4°。由于方位角从正北顺时针测量，正好等于θ，所以方位角约为069°（取整至最接近的度数）。
9. Worked Example: Drawing a Bearing | 例题：绘制方位角
a scale of 1 cm = 2 km, draw the journey and measure the bearing and distance of J from H.
任务：一艘船从H港出发，沿053°方位角航行12公里到达I岛。然后从I岛沿145°方位角航行8公里到达J岛。使用1厘米=2公里的比例尺，画出这段航程，并测量J相对于H的方位角和距离。
Steps: Draw north at H, mark 053° and draw a 6 cm line to I. At I, draw a new north line, mark 145° and draw a 4 cm line to J. Join H to J. Measure the bearing from H to J by placing a protractor at H, aligning 0° with north and reading the angle clockwise to line HJ. Measure HJ in cm and convert to km.
步骤：在H处画出正北方向，标出053°，画一条6厘米的线段到I。在I处重新画出正北线，标出145°，画一条4厘米的线段到J。连接H和J。将量角器放在H处，0°对准正北，顺时针测量直线HJ的角度得到方位角。测量HJ的长度（厘米）并换算为公里。
10. Common Mistakes to Avoid | 常见错误与避免方法
Mistake 1: Forgetting to start from north. Some students measure angles from the wrong reference line, often the vertical grid line that is not labelled north. Always draw a clear north arrow and measure from it.
错误1：忘记从正北开始测量。有些学生从错误的参考线测量角度，常常是未标有正北的垂直网格线。务必画出清晰的正北箭头并从它开始测量。
Mistake 2: Measuring anticlockwise. Bearings are always clockwise. Double‑check the direction of your protractor.
错误2：逆时针测量。方位角始终是顺时针方向。请反复检查量角器的方向。
Mistake 3: Writing a bearing as a one‑ or two‑digit number. Always use three figures. Write 008°, not 8°.
错误3：将方位角写成一或两位数。务必使用三位数字。写作008°，而不是8°。
Mistake 4: Using the back bearing incorrectly. Do not subtract 180° from a bearing less than 180°; always add 180° first and only subtract if the total exceeds 360°.
错误4：反向方位角使用不当。小于180°的方位角不能减去180°；始终先加180°，只有当总和超过360°时才减去180°。
11. Practice Exercise (from Page 180) | 练习（第180页）
Here is a typical problem similar to Exercise 1 on page 180 of your learner book.
以下是一道与教材第180页练习1类似的典型题目。
uct a scale drawing and determine: (a) the bearing of C from A, (b) the distance AC in km.
问题：一架飞机从A机场沿065°方位角飞行300公里到达B点。然后从B点沿140°方位角飞行200公里到达C点。使用1厘米=50公里的比例尺绘制比例图，并确定：(a) C相对于A的方位角，(b) A到C的距离（公里）。
Try to solve this yourself. Remember to draw accurate north lines at both A and B, use your ruler and protractor precisely, and then measure the final leg. Check your back bearing as a self‑assessment: what is the bearing of A from C?
试着自己解答。记住在A和B两点都准确画出正北线，精确使用直尺和量角器，然后测量最终的一段航程。作为自我检测，请计算C相对于A的反向方位角是多少。
12. Summary | 总结
Bearings are three‑figure angles measured clockwise from north. They are essential for describing directions in navigation and for solving scale‑drawing problems. Always write bearings with three digits, use a north line at each new point, and add or subtract 180° to find back bearings. Master these skills and you will be able to tackle any bearing question on your Cambridge assessments with confidence.
方位角是从正北方向顺时针测量的三位数角度，对于描述导航中的方向和解决比例图问题至关重要。始终用三位数字书写方位角，在每一个新点处画出正北线，并通过加或减180°求出反向方位角。掌握这些技巧后，你将能够信心十足地应对剑桥考试中的任何方位角题目。
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