📚 Bearings | 方位角
Bearings are a key topic in KS3 mathematics that link angles, geometry and real‑world navigation. Understanding how to measure, draw and interpret bearings helps you describe directions accurately and solve practical problems. This article takes you through everything you need to know, with clear examples and tips.
方位角是 KS3 数学中的一个重要主题，它将角度、几何与现实世界中的导航联系在一起。理解如何测量、绘制和解读方位角能帮助你准确地描述方向并解决实际问题。本文将带你了解所有你需要知道的内容，并配有清晰的示例和技巧。
1. What are Bearings? | 什么是方位角？
In mathematics and geography, a bearing is an angle that tells you the direction of one place from another. It is always measured from the north direction, moving clockwise.
在数学和地理中，方位角是表示一个地方相对于另一个地方方向的角。它总是从北方向开始，顺时针测量。
Bearings are used by ships, aircraft and walkers to navigate. They are given in degrees, ranging from 0° to 360°.
方位角被船舶、飞机和徒步者用于导航。它们以度为单位，范围从 0° 到 360°。
A bearing of 0° or 360° points directly north. A bearing of 090° points east, 180° points south, and 270° points west.
0° 或 360° 的方位角指向正北。090° 的方位角指向东，180° 指向南，270° 指向西。
2. Compass Directions and Bearings | 罗盘方向与方位角
Before using three‑figure bearings, it helps to understand compass points. The four cardinal directions are north (N), east (E), south (S) and west (W).
在使用三位数方位角之前，了解罗盘方向很有帮助。基本方向有四个：北 (N)、东 (E)、南 (S) 和西 (W)。
Between these are the intercardinal directions: northeast (NE), southeast (SE), southwest (SW) and northwest (NW). Each of these is exactly 45° from the nearest cardinal point.
在这些方向之间还有中间方向：东北 (NE)、东南 (SE)、西南 (SW) 和西北 (NW)。每个都与最近的基本方向相差正好 45°。
We can convert a compass direction into a three‑figure bearing. For example, NE is 045°, SE is 135°, SW is 225°, and NW is 315°.
我们可以将罗盘方向转换为三位数方位角。例如，东北是 045°，东南是 135°，西南是 225°，西北是 315°。
When giving a bearing, we always use three digits. So east is written as 090°, not 90°.
给出方位角时，我们总是使用三位数字。所以东写作 090°，而不是 90°。
3. Three-Figure Bearings | 三位数方位角
A bearing must always be written with three figures. If the angle is less than 100°, we add a leading zero. For instance, a bearing of 47° is written as 047°.
方位角必须总是用三位数字书写。如果角度小于 100°，我们加上前导零。例如，47° 的方位角写作 047°。
Using three figures avoids confusion. It is clear that 005° means 5°, while 050° means 50°. This is especially important in navigation and when plotting routes.
使用三位数可以避免混淆。005° 清楚地表示 5°，而 050° 表示 50°。这在导航和绘制路线时尤其重要。
Angles are measured from north clockwise. So 000° is north, 090° is east, 180° is south, 270° is west, and 360° brings you back to north.
角度从北开始顺时针测量。因此 000° 是北，090° 是东，180° 是南，270° 是西，360° 又回到北。
4. Measuring a Bearing on a Diagram | 在图上测量方位角
To measure the bearing of point B from point A on a diagram, place your protractor with its centre at A and the 0° line pointing north. Read the angle clockwise from north to the line AB.
要在图上测量点 B 相对于点 A 的方位角，将量角器的中心放在 A 点，0° 线指向北。顺时针阅读从北到线段 AB 的角度。
If the line AB points directly east, the bearing reads 090°. If it points southwest, the bearing will be 225°.
如果线段 AB 指向正东，方位角就是 090°。如果指向西南，方位角将是 225°。
Always ensure your protractor is correctly aligned with the north line. Many diagrams include a north arrow; if not, draw one yourself parallel to the grid.
务必确保量角器与北线正确对齐。许多图上标有指北箭头；如果没有，请自己画一条与网格平行的北线。
Practice measuring bearings from simple points so you can quickly identify whether the angle is acute, obtuse or reflex.
通过练习测量简单点的方位角，你可以快速判断角度是锐角、钝角还是优角。
5. Drawing a Bearing and Distance | 绘制方位角和距离
To draw a point when given its bearing and distance from a reference point, first draw a north line at the reference point. Place the protractor and mark the required angle clockwise from north. Draw a line in that direction.
给定某点相对于参考点的方位角和距离，要画出该点，首先在参考点画一条北线。放置量角器，从北顺时针标记所需角度，沿着该方向画一条线段。
Then use the given scale to mark the correct distance along the line. For example, if 1 cm represents 1 km and the distance is 5 km, measure 5 cm along the line.
然后根据给定的比例尺，在线段上标出正确的距离。例如，如果 1 厘米代表 1 千米，而距离是 5 千米，则在线段上量取 5 厘米。
Label the point with a capital letter and write the bearing and distance beside the line. This makes your diagram easy to read.
用大写字母标记该点，并在线段旁写上方位角和距离。这样使你的图易于阅读。
Practice drawing bearings of 060°, 120°, 200° and 300° to become comfortable with the protractor in all four quadrants.
练习绘制 060°、120°、200° 和 300° 的方位角，以便熟练使用量角器处理所有四个象限。
6. Bearings Between Two Points | 两点之间的方位角
The bearing of B from A is measured at A, north clockwise to the line AB. The bearing of A from B is measured at B, north clockwise to the line BA.
B 相对于 A 的方位角是在 A 点测量的，从北顺时针到线段 AB。A 相对于 B 的方位角是在 B 点测量的，从北顺时针到线段 BA。
These two bearings are not the same. In fact, they differ by 180° if both points lie on a straight line. This is because the direction from A to B is opposite to the direction from B to A.
这两个方位角并不相同。事实上，如果两点在一条直线上，它们相差 180°。这是因为从 A 到 B 的方向与从 B 到 A 的方向相反。
However, when the points are not aligned north‑south or east‑ then the bearing of A from B is either x° + 180° (if x < 180°) or x° − 180° (if x ≥ 180°).
但是，当两点不沿南北或东西对齐时，关系不只是在完整圆周上简单加减 180°。规则是：如果 B 相对于 A 的方位角是 x°，那么 A 相对于 B 的方位角是 x° + 180°（若 x < 180°）或 x° − 180°（若 x ≥ 180°）。
This is called a back bearing or reverse bearing. We will look at it more in the next section.
这叫做反向方位角或回程方位角。我们将在下一节详细讨论。
7. Back Bearings (Return Bearings) | 反向方位角（回程方位角）
ring of, say, 110°, then to return from B to A you follow a bearing of 110° + 180° = 290°.
反向方位角是你要返回起点时会采取的方向。如果你从 A 到 B 按方位角 110° 行进，那么从 B 返回 A 的方位角是 110° + 180° = 290°。
g is 250°, then the back bearing is 250° − 180° = 070° (always expressed as three digits).
如果前向方位角是 250°，则反向方位角是 250° − 180° = 070°（总是表示为三位数）。
Remember that bearings must stay between 000° and 360°. If your calculation gives a number greater than 360°, subtract 360°. If it gives a negative number, add 360°.
请记住方位角必须保持在 000° 到 360° 之间。如果你的计算结果大于 360°，减去 360°；如果是负数，则加上 360°。
Understanding back bearings is essential for navigation tasks where you need to plot a return journey.
理解反向方位角对需要绘制回程路线的导航任务至关重要。
8. Bearings and Scale Drawings | 方位角与比例尺图
Bearings are often combined with distances to create scale drawings. A scale drawing allows you to represent a real‑life journey on paper using a smaller scale, such as 1 cm : 1 km.
方位角经常与距离结合来创建比例尺图。比例尺图让您能够用较小的比例（例如 1 cm : 1 km）在纸上表示现实中的行程。
To construct a scale drawing of a journey with several legs, start at the first point. Draw the north line, measure the first bearing, and draw the first leg to the scaled length. At the end of that leg, draw a new north line, measure the next bearing, and repeat.
要绘制一段多段行程的比例尺图，从第一个点开始。画北线，测量第一个方位角，按比例长度画出第一段。在该段末端，画新的北线，测量下一个方位角，重复操作。
This process is used to find the final position or to determine the direct bearing and distance back to the start. After plotting all legs, simply connect the final point to the starting point, measure the angle from north, and measure the length to find the direct return bearing and distance.
这个过程用于找出最终位置，或确定直接返回起点的方位角和距离。绘制完所有段后，只需连接终点和起点，测量从北方向的角度，并测量长度即可得到直接返回的方位角和距离。
Accurate drawing and careful use of a sharp pencil, a ruler and a protractor are key to getting reliable results.
准确的绘图以及仔细使用尖头铅笔、直尺和量角器是获得可靠结果的关键。
9. Using Bearings to Find a Position | 利用方位角确定位置
If you are given the bearing of an unknown point from two known points, you can find its location by drawing the two bearing lines. The point where the lines intersect is the position of the unknown object.
如果你被告知一个未知点相对于两个已知点的方位角，你可以通过画出这两条方位线来找到它的位置。两条线的交点就是未知物体的位置。
This technique, known as triangulation, is used in map reading and search‑and‑rescue operations. For example, a lighthouse is spotted on a bearing of
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