Bearings | 方位角

📚 Bearings | 方位角

Bearings are a way of describing direction using angles measured clockwise from North. They are essential in navigation, mapping, and real-world geometry. In this article, you will learn how to measure, draw, and calculate bearings, covering the key skills needed for KS3 Cambridge Mathematics. We will follow the three golden rules, use protractors accurately, and solve problems involving maps and scale drawings.

方位角是一种通过从正北顺时针测量的角度来描述方向的方法。它们对导航、地图绘制和现实中的几何问题至关重要。在本文中，你将学习如何测量、绘制和计算方位角，掌握 KS3 剑桥数学所需的核心技能。我们将遵循三条黄金法则，准确使用量角器，并解决涉及地图和比例绘图的问题。

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1. What Is a Bearing? | 什么是方位角？

A bearing is an angle measured in degrees, always starting from the North line (true North) and turning clockwise. For example, an object due East has a bearing of 090°. An object due South has a bearing of 180°, and West is 270°. Bearings are always written with three digits, so we write 045° instead of 45°, and 008° for 8°.

方位角是以度为单位测量的角度，总是从正北线（真北）开始并沿顺时针方向转动。例如，正东方的物体方位角为 090°，正南方为 180°，正西方为 270°。方位角始终用三位数字书写，因此 45° 要写成 045°，8° 要写成 008°。

Think of a bearing like a compass reading: you face North, then rotate clockwise until you are looking directly at your target. The amount you turned is the bearing. It removes the confusion of compass points like northeast or southwest and gives a precise number anyone can follow.

你可以把方位角想象成罗盘读数：先面向正北，然后顺时针旋转，直到直接对准目标。你转过的角度就是方位角。它消除了东北、西南等罗盘点的模糊性，给出了任何人均可遵循的精确数字。

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2. The Three Golden Rules of Bearings | 方位角的三条黄金法则

Rule 1: All bearings are measured from the North direction. The starting line is always a vertical line pointing up on a diagram, labelled N.

规则 1：所有方位角都从正北方向开始测量。起始线在图上永远是一条指向上方的垂直线，标注为 N。

Rule 2: Bearings are measured in a clockwise direction. You must go the long way round from North through East, South, West and back to North if needed, always keeping the angle less than 360°.

规则 2：方位角按顺时针方向测量。你必须从北出发，经过东、南、西，再回到北（必要时），始终保持角度小于 360°。

Rule 3: Bearings are always given as three-figure numbers. For example, the bearing of North itself is 000° (or 360°), East is 090°, Southwest is 225°. Never write a bearing as 45° or 7° – always pad with zeros to make three digits.

规则 3：方位角始终用三位数表示。例如，正北本身的方位角是 000°（或 360°），正东是 090°，西南是 225°。切勿将方位角写成 45° 或 7°——务必用零补足三位数字。

These rules eliminate ambiguity. A ship captain reading ‘045°’ knows exactly to turn 45° clockwise from North, while ’45°’ could be mistaken for another measurement. The three-figure system is universal in navigation and surveying.

这些规则消除了歧义。读到“045°”的船长会清楚地知道从北顺时针转 45°，而“45°”可能会被误认为是其他测量值。三位数字体系在导航和测量中通用。

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3. Measuring a Bearing with a Protractor | 用量角器测量方位角

To measure the bearing of point B from point A on a diagram, you need a 360° protractor or a semicircular protractor and careful alignment. Follow these steps:

要测量图上从点 A 到点 B 的方位角，你需要一把 360° 量角器或一把半圆量角器并仔细对齐。请按以下步骤操作：

Step 1: Draw a North line at point A. If the diagram already has a North arrow, you may need to draw a parallel North line through A using the same direction.

步骤 1：在点 A 处画一条正北线。如果图上已有指北箭头，你可能需要过点 A 作一条与之平行的正北线。

Step 2: Place the centre of the protractor exactly on point A. Align the 0° mark (or the baseline of the protractor) with the North line. For a semicircular protractor, ensure the straight edge lies along the North line with the centre at A.

步骤 2：将量角器的中心精确对准点 A。将 0° 刻度（或量角器的基线）与正北线对齐。对于半圆量角器，要确保直边贴着正北线，中心落在点 A 上。

Step 3: Look at the line joining A to B. Read clockwise from 0° until you hit that line. With a 360° protractor, read the outer scale directly. With a semicircular protractor, you may need to add 180° if the angle exceeds 180°.

步骤 3：观察连接 A 与 B 的直线。从 0° 开始顺时针读数，直到该直线为止。使用 360° 量角器时，直接读外圈刻度。使用半圆量角器时，如果角度超过 180°，可能需要加上 180°。

Step 4: Write the angle as a three-digit bearing. For example, if you measure 117°, write it as 117°, but if the angle is less than 100°, pad with leading zeros, e.g., 042°.

步骤 4：将角度写成三位数的方位角。例如，如果你测量出 117°，就写成 117°；但如果角度小于 100°，则用前导零补足，如 042°。

Always double-check you measured clockwise. A common mistake is to measure the smaller anticlockwise angle; bear in mind the rules require clockwise from North.

务必仔细检查是否沿顺时针方向测量。常见错误是测量了较小的逆时针角；请记住，规则要求从正北顺时针测量。

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4. Drawing a Bearing | 绘制方位角

When given a bearing and a distance, you can draw the path on paper. Imagine you are told: ‘From point P, walk 5 cm on a bearing of 130°.’

当给定一个方位角和一段距离时，你可以在纸上绘制路径。假设题目要求：“从点 P 出发，沿 130° 的方位角走 5 厘米。”

Step 1: Mark point P and draw a North line straight upward from it. Use a ruler to keep the North line vertical and label it N.

步骤 1：标记点 P，并从该点向上画一条正北线。用直尺保持正北线垂直，并标注 N。

Step 2: Place your protractor centre on P with 0° pointing North. Find 130° on the protractor scale and make a small mark on the paper at the edge of the protractor.

步骤 2：将量角器中心放在点 P 上，使 0° 指向正北。在量角器刻度上找到 130°，并在量角器边缘的纸上做一个标记。

Step 3: Remove the protractor and draw a line from P through that mark. This line is the required direction.

步骤 3：移开量角器，从点 P 出发穿过该标记画一条直线。这条线就是所需的方向。

Step 4: Using a ruler, measure 5 cm along the line from P to locate the new point, say Q. Label Q and the bearing. You have successfully drawn a bearing of 130°.

步骤 4：用直尺在这条线上从点 P 量出 5 厘米，得到新点，假设为 Q。标注 Q 和方位角。你已成功绘制了 130° 的方位角。

Drawing bearings accurately is important for scale diagrams in map work. Use sharp pencils and always check your protractor alignment.

准确绘制方位角对地图工作中的比例图至关重要。请使用尖铅笔，并始终检查量角器的对齐情况。

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5. Bearings from One Point to Another | 从一点到另一点的方位角

In many problems, you need to find the bearing of one point from another. The bearing of B from A means you stand at A, draw a North line, and find the clockwise angle to line AB. This is often written as ‘bearing of B from A’.

在许多问题中，你需要求从一点看另一点的方位角。“从 A 看 B 的方位角”意味着你站在点 A，画一条正北线，然后找到与线段 AB 之间的顺时针角度。它通常记作“从 A 到 B 的方位角”。

If you are asked for the bearing of A from B, the situation is reversed: you now stand at B, draw a North line, and measure clockwise to line BA. These two bearings are usually different unless the points lie on a North–South line.

如果要求“从 B 看 A 的方位角”，情况则相反：你站在点 B，画一条正北线，然后测量到线段 BA 的顺时针角度。除非两点恰好位于南北线上，否则这两个方位角通常不同。

Here is a table of bearings for common positions of B relative to A:

下面是一张表格，显示了点 B 相对于点 A 位于常见位置时的方位角：

Position of B relative to A	Bearing
Due North	000°
Due East	090°
Due South	180°
Due West	270°
Northeast exactly	045°
Southeast exactly	135°
Southwest exactly	225°
Northwest exactly	315°

Always label the North line at the point you are measuring from, not the target point. This is a very common trap in exams.

测量时，始终在你所在的位置（出发点）画正北线，而不是在目标点。这是考试中非常常见的陷阱。

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6. The Back Bearing (Return Bearing) | 返程方位角

A back bearing is the bearing you would take to return from point B to point A. It is the reverse direction of the forward bearing. To calculate a back bearing:

返程方位角是你从点 B 返回点 A 所需的方位角。它是前向方位角的反方向。计算返程方位角的方法如下：

If forward bearing < 180°, back bearing = forward bearing + 180°

如果前向方位角 < 180°，返程方位角 = 前向方位角 + 180°

If forward bearing > 180°, back bearing = forward bearing – 180°

如果前向方位角 > 180°，返程方位角 = 前向方位角 – 180°

For example, the bearing of B from A is 060°. The back bearing from B to A is 060° + 180° = 240°. If the bearing of Q from P is 210°, then the back bearing is 210° – 180° = 030°.

例如，从 A 到 B 的方位角是 060°，那么从 B 返回 A 的返程方位角是 060° + 180° = 240°。如果从 P 到 Q 的方位角是 210°，则返程方位角为 210° – 180° = 030°。

This rule works because turning exactly 180° points you in the opposite direction along a straight line. If your calculated back bearing ends up as 360°, write it as 000°. For example, 180° + 180° = 360°, which is written 000°.

这条规则之所以成立，是因为恰好转 180° 会让你沿着同一直线的反方向前进。如果你计算出的返程方位角恰好是 360°，那么应写成 000°。例如，180° + 180° = 360°，应记为 000°。

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7. Bearings with Scale Drawings | 含比例绘图的方位角

In many KS3 problems, distances are combined with bearings to create scale drawings. A typical question might state: ‘A lighthouse is 8 km from a port on a bearing of 155°. Draw this using a scale of 1 cm to 2 km.’

在 KS3 的许多问题中，距离与方位角结合使用来绘制比例图。一道典型的题目可能是：“一座灯塔位于港口方位角 155° 的 8 公里处。请用 1 厘米代表 2 公里的比例绘制。”

First convert the real distance to map distance: 8 km becomes 4 cm on the diagram. Draw the port, a North line, measure 155° clockwise, and mark a point 4 cm away. Label clearly with the bearing and the true distance.

首先将实际距离转换为图上的距离：8 公里变为 4 厘米。画出港口和正北线，顺时针量出 155°，并在 4 厘米处标记一点。清晰地标注方位角和实际距离。

Sometimes you are given a diagram with two legs and must find a bearing from the end point back to the start. Use your ruler to draw lines, measure the angle at the appropriate point, and apply the back bearing rule if needed.

有时你会看到一幅包含两段路径的图，并需要求出从终点返回起点的方位角。用直尺画线，在适当的点测量角度，并在需要时应用返程方位角规则。

Accuracy in scale drawings is vital. A small error in angle measurement can lead to a big difference in position over a long distance. Always use a sharp pencil and check that your North lines are parallel.

比例绘图的准确性至关重要。角度测量的微小误差在长距离下可能会导致位置出现很大偏差。务必使用削尖的铅笔，并检查各处的正北线是否相互平行。

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8. Problem Solving with Bearings | 方位角问题求解

Here is a typical multi-step problem: ‘From point X, a tower is 12 km away on a bearing of 040°. From the same point X, a beacon is 15 km away on a bearing of 130°. Find the bearing of the beacon from the tower.’

下面是一道典型的多步骤问题：“从点 X 出发，一座塔位于方位角 040°、12 公里处。从同一点 X 出发，一座信标位于方位角 130°、15 公里处。求从塔看向信标的方位角。”

To solve, draw a scale diagram. Plot X, draw North, mark the tower and beacon positions accurately using the given bearings and the same scale. Then draw a North line at the tower, connect the tower to the beacon, and measure the clockwise angle from the tower’s North line to that line. That angle is the required bearing.

要解答此题，需绘制比例图。画出点 X 及正北线，按照给定的方位角并采用相同比例，准确标出塔和信标的位置。然后在塔的位置画一条正北线，连接塔与信标，测量从塔的正北线出发顺时针转到该连线的角度。这个角度就是所求的方位角。

You may also use angle facts to calculate bearings without drawing. For instance, if you know the angles inside a triangle formed by three points, you can combine them with the known North direction to deduce bearings.

你也可以利用角度关系，不靠绘图直接计算方位角。例如，若已知三点构成的三角形内角，你可以将其与已知的正北方向结合起来推算出方位角。

In exams, draw clear diagrams and label all known bearings and distances. This helps you avoid confusion and shows the examiner your reasoning.

在考试中，应画出清晰的示意图，并标注所有已知的方位角和距离。这有助于避免混乱，并向考官展示你的推理过程。

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9. Real-Life Uses of Bearings | 方位角的实际应用

Bearings are used by pilots, ship captains, hikers, and surveyors. Pilots navigate using a compass and bearing to stay on course. In orienteering, you follow a bearing through the forest to reach the next checkpoint. Surveyors measure bearings between landmarks to map out land boundaries.

飞行员、船长、远足者和测量员都会使用方位角。飞行员依靠罗盘和方位角沿航线飞行。在定向越野中，你遵循方位角穿越森林到达下一个检查点。测量员测量地标之间的方位角以绘制地块边界。

Understanding bearings also helps in technology: GPS systems internally calculate directions using bearings, but they present user-friendly turn-by-turn instructions. Even in video games, non-player characters often navigate using bearing-like pathfinding.

理解方位角对技术也有帮助：GPS 系统在内部使用方位角来计算方向，不过它们会向用户呈现易于理解的逐向导航指令。即使在电子游戏中，非玩家角色也常通过类似方位角的寻路算法来移动。

Mastery of bearings gives you a foundation for further geometry topics like vectors and trigonometry, where direction and angle are essential.

掌握方位角为你学习向量和三角学等更进一步的几何专题打下基础，其中方向与角度不可或缺。

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10. Common Mistakes and Tips | 常见错误与提示

Mistake 1: Measuring anticlockwise from North. Always measure the clockwise angle, which may be larger than 180° but less than 360°.

错误 1：从正北逆时针测量。务必测量顺时针角度，它可能大于 180° 但小于 360°。

Mistake 2: Forgetting the three-digit rule. Write 005° instead of 5°. Examiners often deduct marks for missing leading zeros.

错误 2：忘记三位数字规则。应写 005° 而非 5°。考官常因缺少前导零而扣分。

Mistake 3: Drawing the North line at the wrong point. The North line must be drawn where you are measuring from. When finding the bearing of C from D, the North line starts at D.

错误 3：在错误的点画正北线。正北线必须画在测量的出发点。求“从 D 看 C 的方位角”时，正北线要从点 D 开始画。

Mistake 4: Confusing the forward and back bearing. Remember the ±180° rule and always check which bearing you need: ‘from A’ means stand at A.

错误 4：混淆前向方位角与返程方位角。牢记 ±180° 规则，并始终确认你需要的是哪一个方位角：“从 A”意味着你站在点 A 的位置。

Tip: Underline the words ‘from’ and ‘of’ in the question. This small habit can save you from mixing up directions.

提示：在题目中把“从”和“的”等关键字圈出来。这个小习惯能帮你避免弄混方向。

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11. Quick Practice | 小试牛刀

Try these without looking at the answers immediately. (Answers are at the end of this section.)

尝试回答以下问题，不要立刻看答案。（答案在本节末尾。）

Q1: What is the three-figure bearing for Southwest?

问题 1：西南方向的三位数方位角是多少？

Q2: A ship sails from harbour H on a bearing of 075° for 6 km to a buoy B. What is the bearing of the harbour from the buoy?

问题 2：一艘船从港口 H 出发，沿 075° 方位角航行 6 公里到达浮标 B。从浮标看，港口的方位角是多少？

Q3: On a diagram, point A is exactly north of point B. What is the bearing of A from B, and the bearing of B from A?

问题 3：在图上，点 A 位于点 B 的正北方。从 B 看 A 的方位角是多少？从 A 看 B 的方位角又是多少？

Q4: Using a protractor, a student measures the angle from North to a line as 225° anticlockwise. What is the correct bearing?

问题 4：一名学生用量角器从正北逆时针量得一个角度为 225°。正确的方位角是多少？

Answers: Q1 225°; Q2: 075° + 180° = 255°; Q3: Bearing of A from B = 000°, bearing of B from A = 180°; Q4: 360° – 225° = 135°.

答案：题1 225°；题2 075° + 180° = 255°；题3 从 B 看 A 为 000°，从 A 看 B 为 180°；题4

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