📚 Bearings in Navigation | 导航中的方位角
Bearings are a way of describing direction using angles measured clockwise from North. They are widely used in navigation, map reading, and real‑life problems to give precise directions. In this article, you will learn what bearings are, how to measure and draw them, and how to solve problems involving bearings at Key Stage 3 level.
方位角是一种用从正北方向顺时针测量的角度来描述方向的方法。它们广泛应用于导航、地图阅读和实际问题中，以提供精确的方向指引。在本文中，你将学习什么是方位角，如何测量和绘制它们，以及如何解决 KS3 阶段涉及方位角的问题。
1. What Is a Bearing? | 什么是方位角？
A bearing is an angle, measured in degrees, that tells you the direction of one point from another. The angle is always measured clockwise from the North direction (000°). Bearings are always written as three‑digit numbers, for example 045°, 120°, or 005°.
方位角是一个以度为单位的角度，告诉你一个点相对于另一个点的方向。这个角度总是从正北方向（000°）顺时针测量。方位角始终用三位数字表示，例如 045°、120° 或 005°。
The main rules for bearings are:
• Measure clockwise from North.
• Always give the bearing as a three‑figure number (use leading zeros if needed).
• The bearing tells you the direction you would face to look from the starting point to the target point.
方位角的主要规则是：
• 从正北方向顺时针测量。
• 始终用三位数字表示方位角（如果需要，前面加零）。
• 方位角告诉你从起点看向目标点所需面对的方向。
2. The Compass Directions | 罗盘方位
Before we use bearings, we recall the main compass points. North is 000° (or 360°), East is 090°, South is 180°, and West is 270°. These are the four cardinal directions. Halfway between these are the intercardinal directions: Northeast (045°), Southeast (135°), Southwest (225°), and Northwest (315°).
在使用方位角之前，我们先回顾一下主要的罗盘方向。北是 000°（或 360°），东是 090°，南是 180°，西是 270°。这是四个基本方向。在这些方向之间的是中间方向：东北（045°）、东南（135°）、西南（225°）和西北（315°）。
| Direction | Bearing |
| North | 000° |
| Northeast | 045° |
| East | 090° |
| Southeast | 135° |
| South | 180° |
| Southwest | 225° |
| West | 270° |
| Northwest | 315° |
3. Measuring a Bearing | 测量方位角
To find the bearing of point B from point A, imagine standing at A facing North. Then turn clockwise until you are facing directly towards B. The angle you have turned through is the bearing of B from A.
要找出点 B 相对于点 A 的方位角，想象自己站在 A 点面朝正北。然后顺时针转动，直到你正对着 B 点。你所转过的角度就是 B 相对于 A 的方位角。
On a diagram, draw a North line at the starting point A. Join A to B with a straight line. Use a protractor to measure the clockwise angle from the North line to the line AB. If the angle is less than 100°, add leading zeros to make it three figures.
在图上，在起点 A 处画一条指向北的线。用一条直线连接 A 和 B。用量角器测量从正北线到 AB 线的顺时针角度。如果角度小于 100°，加上前导零使其成为三位数。
4. Drawing a Bearing | 绘制方位角
To draw a point on a bearing, start by plotting the starting point. Draw a North line vertically upward from that point. Place your protractor with 0° along the North line, then mark the required angle clockwise. Draw a line from the starting point through your mark. This line shows the direction of the bearing. You can then use a scale to mark the distance to the target point.
要按方位角绘制一个点，首先标出起点。从该点垂直向上画一条正北线。将量角器的 0° 刻度线对准正北线，然后顺时针标记所需的角度。从起点出发，经过标记点画一条直线。这条线显示了方位角的方向。然后你可以用比例尺标出到目标点的距离。
5. The Three-Figure Rule | 三位数规则
Bearings must always be written with exactly three digits. For example, a bearing of 60° is written as 060°. A bearing of 8° becomes 008°. This rule avoids confusion between numbers like 45° and 450°, and it is essential when communicating directions in navigation and aviation.
方位角必须始终用恰好三位数字书写。例如，60° 的方位角应写为 060°。8° 的方位角变为 008°。这条规则避免了像 45° 和 450° 这样的数字之间的混淆，在导航和航空中传达方向时至关重要。
6. Back Bearings | 反方位角
The back bearing is the bearing you would use to return from point B to point A. It is the opposite direction. To find a back bearing, add 180° to the original bearing if it is less than 180°. If the original bearing is 180° or more, subtract 180°. The result should always be a three‑figure number.
反方位角是从点 B 返回点 A 所使用的方位角。它是相反的方向。要找到反方位角，如果原方位角小于 180°，则加上 180°；如果原方位角大于或等于 180°，则减去 180°。结果应该始终是一个三位数。
Example: The bearing of B from A is 075°. The back bearing (bearing of A from B) is 075° + 180° = 255°. Example 2: Bearing of C from D is 210°. Back bearing = 210° – 180° = 030°.
示例：B 相对于 A 的方位角是 075°。反方位角（A 相对于 B 的方位角）是 075° + 180° = 255°。示例 2：C 相对于 D 的方位角是 210°。反方位角 = 210° – 180° = 030°。
7. Problems with Several Points | 多点方位问题
In KS3 problems, you may be given bearings and distances between several points. For example, a ship sails from port P on a bearing of 120° for 8 km to point Q, then on a bearing of 210° for 5 km to point R. You need to draw a scale diagram to find the final position or the bearing and distance back to the start.
在 KS3 的问题中，你可能会遇到多个点之间的方位角和距离。例如，一艘船从港口 P 出发，沿 120° 方位角航行 8 公里到达点 Q，然后沿 210° 方位角航行 5 公里到达点 R。你需要绘制一个比例图来找到最终位置，或者返回起点的方位角和距离。
Always draw a new North line at each new point before measuring the next bearing. This is a common mistake: students often measure all angles from one North line, but bearings are always measured from the North line at the point you are starting from.
在测量下一个方位角之前，始终在每个新点处绘制一条新的正北线。这是一个常见错误：学生常常从一条正北线测量所有角度，但方位角始终是从你所在的起点处的正北线开始测量的。
8. Using Scale Drawings with Bearings | 使用比例尺绘图与方位角
Scale drawings allow you to solve navigation problems without using trigonometry. Choose a suitable scale, for example 1 cm represents 1 km. Draw the North lines, measure the bearings accurately with a protractor, and use a ruler to mark the distances. At the end, you can measure the final bearing and distance directly from the diagram.
比例尺绘图可以让你在不使用三角函数的情况下解决导航问题。选择一个合适的比例，例如 1 厘米代表 1 千米。画出正北线，用量角器精确测量方位角，然后用直尺标出距离。最后，你可以直接从图中测量出最终的方位角和距离。
Remember: accuracy is very important. A small error in measuring a bearing can lead to a large error in the final position over long distances.
记住：准确性非常重要。在测量方位角时的一个小错误，在长距离上可能导致最终位置的巨大误差。
9. Bearings and Angle Facts | 方位角与角度关系
Bearings are closely linked to angles on a straight line, around a point, and in triangles. Since North lines are parallel, you can use alternate angles and corresponding angles to find unknown bearings without a protractor. For instance, if you know the bearing of B from A is x°, you might find the interior angle at B using parallel line rules.
方位角与直线上的角、点周围的角以及三角形中的角密切相关。由于正北线是平行的，你可以利用内错角、同位角等角度关系，在不用量角器的情况下找出未知的方位角。例如，如果你知道 B 相对于 A 的方位角是 x°，你可以利用平行线规则求出 B 处的内角。
This skill is especially useful when you are given a diagram and asked to calculate a bearing rather than measure it.
当你看到一幅图并被要求计算而不是测量方位角时，这一技能尤其有用。
10. Common Mistakes to Avoid | 常见错误及避免方法
Many students lose marks by forgetting to write bearings as three figures, by measuring anticlockwise instead of clockwise, or by using the same North line for all points. Always read the question carefully: the bearing of A from B is different from the bearing of B from A.
许多学生因为没有将方位角写成三位数、逆时针测量而不是顺时针测量，或者对所有点使用同一条正北线而丢分。一定要仔细读题：A 相对于 B 的方位角与 B 相对于 A 的方位角是不同的。
Another typical mistake is forgetting to add or subtract 180° when finding a back bearing, or not checking that the result is a three‑figure number. Always double‑check your answers.
另一个典型错误是在求反方位角时忘记加上或减去 180°，或者没有检查结果是否为三位数。务必仔细检查你的答案。
11. Real‑Life Applications | 实际应用
Bearings are used by pilots, ship captains, hikers, and surveyors. Air traffic controllers give bearings to pilots to guide aircraft. Ships use bearings to navigate between ports. Even smartphone map apps use bearing calculations to show your direction of travel.
飞行员、船长、徒步旅行者和测量员都使用方位角。空中交通管制员向飞行员提供方位角以引导飞机。船舶利用方位角在港口之间导航。甚至智能手机地图应用也使用方位角计算来显示你的行进方向。
Understanding bearings helps you read maps, plan routes, and develop spatial awareness. It is a fundamental skill in geometry and geography.
理解方位角有助于你阅读地图、规划路线并培养空间感。这是几何和地理中的一项基本技能。
12. Practice and Mastery | 练习与掌握
To become confident with bearings, practise drawing and measuring them on plain paper and on grid paper. Start with simple questions where points are aligned with compass directions, then move to more complex diagrams with several legs of a journey. Always draw North lines clearly and check your protractor alignment.
要熟练掌握方位角，请在白纸和方格纸上练习绘制和测量它们。从点与罗盘方向对齐的简单问题开始，然后尝试包含多段行程的较复杂图表。始终清晰地画出正北线，并检查量角器的对齐情况。
Remember the golden rules: measure clockwise, three figures, North line at the point you are measuring from. With these principles, you can tackle any KS3 bearing problem successfully.
记住黄金法则：顺时针测量、三位数字、在测量起点处画正北线。掌握这些原则，你就能成功解决任何 KS3 的方位角问题。
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