📚 Mastering Bearings and Scale Drawings | 精通方位角与比例绘图
A bearing is a three-figure angle used to describe direction. It is measured clockwise from the north direction. Scale drawings allow us to represent real distances on paper using a ratio. Both ideas appear throughout the Cambridge KS3 mathematics course, especially in position, movement and map-work problems.
方位角是用来描述方向的三位数角度。它是从北方向按顺时针方向测量的。比例绘图让我们能够用比例在纸上表示实际距离。这两个概念在剑桥 KS3 数学课程中经常出现，尤其是在位置、运动和地图问题中。
1. What Are Bearings? | 什么是方位角？
A bearing is an angle measured clockwise from the north direction. In Cambridge KS3 mathematics, bearings are always written as three-figure angles, such as 045° or 210°. We use bearings to describe the direction of one point from another without relying on vague words like 'left' or 'right'. A full turn is 360°, so every bearing is between 000° and 360°.
方位角是从北方向顺时针测量的角度。在剑桥 KS3 数学中，方位角始终写成三位数角度，例如 045° 或 210°。我们使用方位角描述一个点相对于另一个点的方向，而不依赖 '左' 或 '右' 等模糊词语。一整圈为 360°，因此每个方位角都在 000° 到 360° 之间。
Bearings are widely used in navigation, map reading and scale drawing. A ship's course, an aircraft flight path and a walker's route can all be described by a bearing and a distance. Once you know the bearing and distance, you can plot the movement accurately on paper.
方位角广泛用于导航、地图阅读和比例绘图。船只航线、飞机飞行路径和步行者的路线都可以用方位角和距离来描述。一旦你知道方位角和距离，就可以在纸上准确地画出运动路线。
2. Reading and Writing Three-Figure Bearings | 读与写三位数方位角
When writing a bearing, the angle must always have three digits. If the angle is less than 100°, we add a zero or two zeros at the front. For example, an angle of 45° is written as 045°. An angle of 7° is written as 007°. If the angle is exactly 90°, we write 090°.
书写方位角时，角度必须始终有三位数字。如果角度小于 100°，我们在前面加一个或两个零。例如，45° 应写成 045°。7° 应写成 007°。如果角度正好是 90°，我们写 090°。
The table below shows some common directions and their three-figure bearings.
下表显示了一些常见方向及其三位数方位角。
| Direction | Bearing |
|---|---|
| North | 000° |
| East | 090° |
| South | 180° |
| West | 270° |
| Northeast | 045° |
| Southwest | 225° |
Always check that the bearing is given as a three-figure angle. A common mistake is to write 45° instead of 045° in a bearing question.
务必检查方位角是否写成了三位数角度。一个常见错误是在方位角问题中写 45° 而不是 045°。
3. Rules for Measuring Bearings | 测量方位角的规则
There are four important rules when measuring a bearing. First, always measure from north. Second, always turn clockwise. Third, always write the answer with three figures. Fourth, when you read 'the bearing of A from B', stand at B and face north before measuring the angle to A.
测量方位角时有四条重要规则。第一，始终从北方向开始测量。第二，始终按顺时针方向旋转。第三，答案始终写成三位数。第四，当你读到 'A 相对于 B 的方位角' 时，站在 B 点并面向北方，然后测量到 A 的角度。
For example, to find the bearing of a tree from a house, draw a north line at the house. Place your protractor with 0° pointing north. Measure clockwise from 0° to the line joining the house to the tree. If the angle is 145°, write the bearing as 145°.
例如，要求一棵树相对于一座房子的方位角，请在房子处画一条北方向线。将量角器的 0° 指向北方。从 0° 开始顺时针测量到连接房子和树的线段。如果角度是 145°，则方位角写为 145°。
4. Scale Drawings and Maps | 比例绘图与地图
A scale drawing is a drawing that represents a real object or place using a ratio. A map scale such as 1 : 50 000 means that 1 cm on the map represents 50 000 cm in real life. Since 50 000 cm is 500 m, every 1 cm on the map stands for 500 m on the ground.
比例绘图是使用比例表示真实物体或地点的图。地图比例尺如 1 : 50 000 表示地图上的 1 cm 代表实际中的 50 000 cm。因为 50 000 cm 等于 500 m，所以地图上的每 1 cm 代表地面上的 500 m。
The first number in a scale is usually the length on the drawing or map. The second number is the real length. A scale of 2 : 1 is an enlargement: 2 cm on the drawing represents 1 cm in real life. A scale of 1 : 100 is a reduction: 1 cm on the drawing represents 100 cm in real life.
比例尺中的第一个数字通常是绘图或地图上的长度。第二个数字是实际长度。比例尺 2 : 1 表示放大：图上的 2 cm 代表实际中的 1 cm。比例尺 1 : 100 表示缩小：图上的 1 cm 代表实际中的 100 cm。
5. Choosing a Sensible Scale | 选择合适的比例尺
Before drawing a scale diagram, choose a scale that fits your page. If the real distance is 10 km and your page can fit a 20 cm line, a scale of 1 cm : 0.5 km or 1 cm : 500 m is sensible. Then the 10 km distance will be 20 cm long on the page.
在画比例图之前，选择一个适合页面的比例尺。如果实际距离是 10 km，而你的页面可以容纳 20 cm 长的线段，那么 1 cm : 0.5 km 或 1 cm : 500 m 的比例尺是合理的。这样 10 km 的距离在页面上就是 20 cm 长。
To select a scale, convert all units to the same unit first. It is easier to write 1 cm : 500 m than to mix centimetres and kilometres. You can also use the ratio form: 1 : 50 000 is the same as 1 cm : 50 000 cm.
选择比例尺时，先将所有单位换算成相同单位。写 1 cm : 500 m 比混用厘米和千米更容易。你也可以使用比的形式：1 : 50 000 与 1 cm : 50 000 cm 相同。
6. Converting Between Map Length and Real Length | 地图长度与实际长度互换
The key formula for scale drawing is:
比例绘图的关键公式是：
actual length = map length × scale factor
Here the scale factor is the second number in the scale ratio when the first number is 1. For a scale of 1 : 25 000, multiply a map length in centimetres by 25 000 to get the real length in centimetres. Then convert to metres or kilometres if needed.
这里的比例因子是比例中第一个数字为 1 时的第二个数字。对于 1 : 25 000 的比例尺，将以厘米为单位的地图长度乘以 25 000，就得到以厘米为单位的实际长度。然后根据需要换算为米或千米。
of 4 × 25 000 = 100 000 cm. Since 100 000 cm = 1 000 m = 1 km, the real road is 1 km long.
例如，在 1 : 25 000 地图上一条 4 cm 的道路，实际长度为 4 × 25 000 = 100 000 cm。因为 100 000 cm = 1 000 m = 1 km，所以这条道路实际长 1 km。
To go from real length to map length, divide by the scale factor:
要从实际长度求地图长度，除以比例因子：
map length = actual length ÷ scale factor
Always check units before dividing. Convert the real length into centimetres first if the scale is given in centimetres.
除法前一定要检查单位。如果比例尺以厘米为单位，请先将实际长度换算为厘米。
7. Using a Protractor on a Diagram | 在图中使用量角器
A protractor is used to measure and draw bearings. Place the centre of the protractor exactly on the point where the bearing is measured from. Align the 0° line of the protractor with the north direction. Then read the scale clockwise from 0° to the required angle.
量角器用于测量和绘制方位角。将量角器的中心精确放在测量方位角的点上。将量角器的 0° 线与北方向对齐。然后从 0° 开始顺时针读取到所需的角度。
If you are drawing a bearing, mark the required angle with a small dot or line. Remove the protractor and draw a straight line from the starting point through the mark. Label the angle clearly as a three-figure bearing.
如果要绘制方位角，请用小点或短线标出所需角度。移开量角器，从起点穿过标记画一条直线。清楚地将角度标注为三位数方位角。
When measuring from a diagram, always check which direction is north. If no north line is given, draw one yourself. The north line should be vertical and point upwards from the starting point.
从图中测量时，始终检查哪个方向是北方。如果没有给出北线，请自己画一条。北线应从起点垂直向上。
8. Constructing a Scale Drawing Step by Step | 分步构建比例绘图
Follow these steps to draw a journey using bearings and distances. First, mark the starting point and draw a north line. Second, place the protractor at the starting point and measure the first bearing clockwise from north. Third, draw a line in that direction.
按照以下步骤使用方位角和距离绘制路线。首先，标出起点并画一条北线。第二，将量角器放在起点处，从北方向顺时针测量第一个方位角。第三，沿该方向画一条线。
Fourth, use the chosen scale to convert the first real distance to a drawing length. Measure this length along the line and mark the second point. Fifth, at the second point draw a new north line, because the bearing is measured from north at each new position. Sixth, repeat the process for the next bearing and distance.
第四，使用所选比例尺将第一个实际距离换算为绘图长度。沿该线测量这个长度并标出第二个点。第五，在第二个点画一条新的北线，因为方位角是在每个新位置从北方测量的。第六，对下一个方位角和距离重复此过程。
- Step 1: mark the starting point and draw north
- Step 2: measure the bearing clockwise with a protractor
- Step 3: draw the direction line
- Step 4: use the scale to set the distance
- Step 5: draw a new north line at the new point
- Step 6: repeat for the next stage
- 步骤 1：标出起点并画出北方向
- 步骤 2：用量角器顺时针测量方位角
- 步骤 3：画出方向线
- 步骤 4：使用比例尺设定距离
- 步骤 5：在新点处画一条新的北方向线
- 步骤 6：对下一段重复
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Many students lose marks by making the same avoidable mistakes. The table below lists common bearing and scale drawing errors and the correct method for each one.
许多学生因为同样的可避免错误而失分。下表列出了方位角和比例绘图的常见错误以及每种错误对应的正确方法。
| Mistake | Correct approach |
|---|---|
| Measuring anticlockwise from north | Always measure clockwise from north |
| Writing 45° instead of 045° | Always use three figures |
| Forgetting to draw a north line at each new point | Draw a new north line at every point where a bearing is measured |
| Mixing units such as cm and km | Convert all lengths to the same unit before converting |
| Using the wrong scale factor | Check whether you are going from map to real or real to map |
Before starting a question, read it twice and underline the words 'from', 'bearing' and the scale. This small habit helps you avoid most errors.
开始做题前，请阅读两遍题目，并在 'from'、'bearing' 和比例尺下面划线。这个小习惯可以帮助你避免大多数错误。
10. Exam-Style Practice Ideas | 考试风格练习思路
Try this typical Cambridge KS3 exam-style problem. A ship leaves port P and sails 8 km on a bearing of 060° to point Q. From Q it sails 5 km on a bearing of 150° to point R. Use a scale of 1 cm : 1 km to make an accurate scale drawing. Then measure the distance PR and the bearing of R from P.
试试这道典型的剑桥 KS3 考试风格问题。一艘船从港口 P 出发，沿 060° 方位角航行 8 km 到达 Q 点。从 Q 点它沿 150° 方位角航行 5 km 到达 R 点。使用 1 cm : 1 km 的比例尺绘制精确的比例图。然后测量 PR 的距离以及 R 相对于 P 的方位角。
To solve it, draw P and a north line. Use a protractor to draw 060° from north at P. Since the scale is 1 cm : 1 km, the first line PQ must be 8 cm long. Label Q. At Q draw a new north line and draw 150° from north. Make QR 5 cm long. Label R. Join P to R. Measure PR with a ruler: it should be about 9.4 cm, so the real distance is about 9.4 km. Measure the bearing of R from P: place the protractor at P, align 0° with north and read clockwise. The angle should be about 092°.
解题时，画出 P 点和北线。在 P 点用量角器从北方向画出 060°。因为比例尺为 1 cm : 1 km，第一段 PQ 应为 8 cm 长。标出 Q 点。在 Q 点画一条新的北线，从北方向画出 150°。使 QR 长度为 5 cm。标出 R 点。连接 P 和 R。用直尺测量 PR：应约为 9.4 cm，因此实际距离约为 9.4 km。测量 R 相对于 P 的方位角：将量角器放在 P 点，0° 对准北方并顺时针读数。角度应约为 092°。
This type of question tests both bearing measurement and scale conversion. The two must work together: if the distance scale is wrong, the final bearing and distance will also be wrong.
这类问题同时考查方位角测量和比例换算。两者必须配合使用：如果距离比例错误，最终的方位角和距离也会出错。
11. Summary and Key Formulae | 总结与关键公式
The essential facts for bearings are simple. A bearing is always measured from north, always goes clockwise and is always written with three figures. The bearing of A from B means you stand at B and face north before turning clockwise to A.
方位角的核心事实很简单。方位角总是从北方向开始测量，总是顺时针旋转，并且总是写成三位数。A 相对于 B 的方位角意味着你站在 B 点面向北方，然后顺时针转向 A 点。
For scale drawings, remember the two conversion formulae and use the same units throughout.
对于比例绘图，请记住这两个换算公式，并始终使用相同的单位。
actual length = map length × scale factor
map length = actual length ÷ scale factor
| Concept | Key rule |
|---|---|
| Bearing | Clockwise from north, three figures |
| Scale 1 : n | 1 unit on paper represents n units in real life |
| Map to real | Multiply by scale factor |
| Real to map | Divide by scale factor |
Practise drawing three-stage journeys using different scales and bearings. Check each measurement twice before moving on to the next step.
练习使用不同比例尺和方位角绘制三段路线。进入下一步前，请对每次测量检查两遍。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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