📚 Interpreting Pie Charts | 解读饼图
A pie chart is a circular diagram divided into sectors that represent data. Each sector’s angle and area are proportional to the frequency or percentage it represents. In KS3 Cambridge Mathematics, you will learn how to read data from a pie chart, calculate frequencies, find missing values and even construct your own pie charts. Mastering pie charts helps you interpret survey results, compare parts of a whole and present information clearly.
饼图是一种由若干扇形组成的圆形图表。每个扇形的角度与面积同它所代表的数据值（频数或百分比）成正比。在 KS3 剑桥数学中，你将学习如何从饼图中读取信息、计算频数、求解缺失值，甚至自己绘制饼图。掌握饼图能帮助你解读调查结果、比较整体中的各个部分并清晰地展示信息。
1. What Is a Pie Chart? | 什么是饼图？
A pie chart uses a circle (360°) to show how a total amount is divided. Each slice, or sector, has an angle at the centre and an area that match its share of the total. The whole circle always represents 100 % or the total frequency. For example, if a pie chart shows the favourite fruits of 180 students, the full circle represents all 180 students, and each sector shows how many students chose each fruit.
饼图用一个完整的圆（360°）来表示一个总量是如何划分的。每个扇形（扇区）的圆心角和面积都与它所占的份额对应。整个圆始终代表 100% 或总频数。例如，一张饼图展示了 180 名学生最喜欢的水果，整个圆代表全部 180 名学生，每个扇形则显示选择各水果的学生人数。
The size of a sector is given by the formula:
Angle = (frequency ÷ total frequency) × 360°.
扇区的大小通过以下公式计算：
角度 =（频数 ÷ 总频数）× 360°。
2. Reading Data from a Pie Chart | 从饼图中读取数据
When a pie chart is labelled with frequencies or percentages, you can read the information directly. A sector labelled “30” means 30 items belong to that category. However, many pie charts only give the angles or percentages, so you must calculate the frequencies yourself. Always check the total first: the graph may state the total frequency in the title or you may need to work it out from one known sector.
当饼图上直接标注了频数或百分比时，你可以直接读取信息。标有“30”的扇形意味着该类别有 30 个数据项。但很多饼图只给出角度或百分比，这时需要自己计算频数。务必先确认总数：图表标题可能给出了总频数，或者你需要根据某一个已知的扇形来推算总数。
Example: A pie chart shows favourite sports. The total number of students is 240. The football sector has an angle of 90°. Frequency = (90 ÷ 360) × 240 = 60 students.
示例：一张饼图展示了最喜欢的运动。学生总数为 240。足球扇形的角度是 90°。频数 = (90 ÷ 360) × 240 = 60 名学生。
3. Understanding Angles, Fractions and Percentages | 理解角度、分数和百分比
Every sector angle can be written as a fraction of 360° and as a percentage of 100 %. An angle of 90° is 90/360 = 1/4 = 25 %. An angle of 180° is 1/2 = 50 %. An angle of 45° is 1/8 = 12.5 %. To convert an angle to a percentage, divide the angle by 360 and multiply by 100. To find the angle from a percentage, divide the percentage by 100 and multiply by 360.
每个扇形的角度都可以表示为 360° 的分数，也可以表示为 100% 的百分比。90° 的角是 90/360 = 1/4 = 25%。180° 的角是 1/2 = 50%。45° 的角是 1/8 = 12.5%。要将角度转换为百分比，用角度除以 360 再乘以 100。从百分比求角度，则用百分比除以 100 再乘以 360。
This relationship is the key to all pie chart calculations. If a pie chart has no numbers but you measure an angle of 72° with a protractor, you know that sector represents 72/360 = 1/5 = 20 % of the total.
这一关系是所有饼图计算的关键。如果一张饼图没有任何数字，但你用量角器测出某个角为 72°，就可以知道该扇形代表 72/360 = 1/5 = 总体的 20%。
4. Calculating Frequencies from Angles | 由角度计算频数
To find the frequency of a sector, use:
Frequency = (sector angle ÷ 360°) × total frequency.
要计算某个扇形的频数，使用公式：
频数 =（扇形角度 ÷ 360°）× 总频数。
Suppose a pie chart displays the eye colours of 540 people. The ‘blue’ sector is 100°. The frequency of blue = (100 ÷ 360) × 540 = 150. Always remember to divide by 360 first, then multiply by the total. Round your answer to the nearest whole number if people or objects cannot be split.
假设一张饼图展示了 540 人的眼睛颜色。“蓝色”扇形的角度是 100°。蓝色频数 = (100 ÷ 360) × 540 = 150。始终记住先除以 360，再乘以总数。如果对象是人或不可分割的物体，要把答案四舍五入到整数。
If the total frequency is not given, you can find it using a known sector. For example, if you know there are 40 people in a 60° sector, then total = 40 × (360 ÷ 60) = 240.
如果总频数未知，可以利用一个已知的扇形来求。例如，已知一个 60° 的扇形代表 40 人，则总数 = 40 × (360 ÷ 60) = 240。
5. Working Out Missing Angles | 计算缺失的角度
When a pie chart has one unlabelled sector, you can find its angle by subtracting the sum of the known angles from 360°. The angles in a pie chart must add up to 360° exactly. If you have angles of 150°, 90° and 60°, the missing angle is 360 – (150 + 90 + 60) = 60°.
当饼图中有一个扇区未标注时，可以用 360° 减去已知角度之和来得到它的角度。饼图中所有角度之和必须恰好等于 360°。如果有角度 150°、90° 和 60°，那么缺失的角度就是 360 – (150 + 90 + 60) = 60°。
Then you can find the missing frequency or percentage using the formulas above. Always double-check that the angles sum to 360°, including the one you’ve just calculated.
然后可以用前面的公式求出缺失的频数或百分比。务必验算所有角度（包括刚刚算出的）之和是否等于 360°。
6. Comparing Categories Using a Pie Chart | 用饼图比较类别
A pie chart makes it easy to compare categories visually. The larger the sector, the bigger the share. Even without exact numbers, you can say whether one category is more than half, about a quarter or smaller than another. For instance, if the chocolate sector is 180° and vanilla is 60°, chocolate is three times as popular as vanilla because 180 ÷ 60 = 3.
饼图能让人直观地比较各个类别。扇形越大，份额越高。即使没有确切数字，也能判断某个类别是否超过一半、大约四分之一或小于另一个类别。例如，如果巧克力扇区是 180°，香草是 60°，那么巧克力的人气是香草的 3 倍，因为 180 ÷ 60 = 3。
You can also compare using calculated frequencies or percentages. Answers such as ‘half of the students chose walking’ or ‘the number who prefer cycling is twice the number who prefer driving’ are typical. Always quote the categories and the basis of comparison clearly.
你也可以通过计算出的频数或百分比进行比较。典型的表述包括“一半的学生选择了步行”或“喜欢骑车的人数是喜欢开车人数的两倍”。进行这种比较时，要清楚地指出类别和比较的依据。
7. Drawing Your Own Pie Chart | 绘制自己的饼图
To draw a pie chart by hand, first calculate each sector angle using: Angle = (frequency ÷ total) × 360°. Then use a pair of compasses to draw a circle. Draw a vertical radius straight up from the centre, and use a protractor to measure each angle from the vertical, moving clockwise. Label each sector clearly, or provide a key. Give the chart a title stating ‘Pie chart showing …’ and include the total number.
要徒手绘制饼图，先根据公式 角度 =（频数 ÷ 总数）× 360° 计算出每个扇形的角度。然后用圆规画一个圆。从圆心垂直向上画一条半径，用量角器从这条半径开始顺时针测量每个角度。清晰地标注每个扇形，或使用图例。为图表加上标题，例如“显示……的饼图”，并注明总数。
Example: Favourite pets of 30 children – dog: 12, cat: 9, rabbit: 6, other: 3.
Dog angle: (12/30)×360 = 144°; Cat: (9/30)×360 = 108°; Rabbit: (6/30)×360 = 72°; Other: (3/30)×360 = 36°.
Check: 144+108+72+36 = 360°.
示例：30 个孩子最喜欢的宠物——狗：12，猫：9，兔：6，其他：3。
狗的扇形角：(12/30)×360 = 144°；猫：(9/30)×360 = 108°；兔：(6/30)×360 = 72°；其他：(3/30)×360 = 36°。
验算：144+108+72+36 = 360°。
8. Pie Charts with Percentages Only | 仅含百分比的饼图
If a pie chart only shows percentages, you can still find frequencies if the total is known. Frequency = (percentage ÷ 100) × total. Alternatively, if you know one frequency, you can find the total. For example, a pie chart of transport to school shows 25% walk and you are told 60 students walk. The total number of students is 60 ÷ 0.25 = 240.
如果饼图只显示了百分比，在知道总数的情况下仍可求出频数。频数 =（百分比 ÷ 100）× 总数。反过来，若已知某类的频数，也可以求出总数。例如，一张上学交通方式的饼图显示 25% 的学生步行，并且已知步行人数为 60 人。学生总数为 60 ÷ 0.25 = 240。
You can also compare percentages directly: 40 % is four times 10 %, so the frequency ratio is 4 : 1. Always double-check that all percentages add up to 100 %. If they sum to 99 % or 101 %, a small rounding has occurred – this is acceptable.
也可以直接比较百分比：40% 是 10% 的四倍，因此频数比就是 4∶1。要仔细核对所有百分比之和是否为 100%。如果加起来是 99% 或 101%，说明存在轻微的四舍五入，这是可以接受的。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
One common mistake is forgetting to divide by the total frequency before multiplying by 360 when calculating angles, or doing the opposite when finding frequencies. Always check the order: for angle → (part/whole) × 360; for frequency → (angle/360) × total. Another error is assuming the total frequency is 360 – it isn’t; 360 is the number of degrees in a circle. The total must be stated separately.
常见错误之一是在计算角度时忘记先除以总频数再乘 360，或是在求频数时把顺序弄反了。请记住正确的顺序：求角度 →（部分/整体）× 360；求频数 →（角度/360）× 总数。另一个错误是以为总频数就是 360——并非如此，360 是圆的度数，总数必须另外给出或求出。
Students sometimes misread a bar chart or pictogram as a pie chart and apply the wrong method. Make sure you identify the chart type. Also, when measuring with a protractor, always measure from the same starting radius and double-check that the sum of measured angles is close to 360°.
学生有时会把条形图或象形图误认作饼图而用错方法。一定要先确认图表的类型。此外，用量角器测量时，务必从同一条半径开始量起，并检查所有测量角之和是否接近 360°。
10. Real-life Uses of Pie Charts | 饼图的实际应用
Pie charts are widely used to show budget shares, survey results, demographic data and market share. For example, a family budget pie chart might display 30 % housing, 20 % food, 15 % transport and so on. In school, you might use a pie chart to present the results of a class vote on a charity day activity.
饼图被广泛用于展示预算分配、调查结果、人口统计数据和市场份额。例如，家庭预算饼图可能显示住房占 30%、食品 20%、交通 15% 等。在学校，你可能会用饼图来展示班级对于慈善日活动的投票结果。
Being able to interpret pie charts helps you quickly grasp how a total is split. When you see a pie chart in a news article, you can calculate the actual numbers if the total is provided. This skill is essential for topics such as ratio, proportion and data handling in KS3 and beyond.
会解读饼图可以帮助你快速了解一个整体是如何分配的。当你在新闻文章中看到饼图时，如果提供了总数，你就能计算出实际数字。这项技能对于 KS3 及以后的比和比例、数据处理等课题至关重要。
11. Comparing Two Pie Charts | 比较两张饼图
Sometimes you are given two pie charts for similar data, but the totals may be different. For example, one chart shows favourite colours of 100 Year 7 students, and another shows the same for 180 Year 9 students. A sector that looks the same size may represent different frequencies. To make a fair comparison, you must calculate actual frequencies or compare percentages.
有时题目会提供同一类数据的两张饼图，但总数可能不同。例如，一张图显示 100 名七年级学生最喜欢的颜色，另一张显示 180 名九年级学生的同样信息。看起来大小相同的扇形可能代表不同的频数。要进行公正的比较，必须计算实际频数或比较百分比。
Statements like ‘A higher proportion of Year 9 students chose blue’ are valid. Avoid saying ‘more Year 9 students like blue’ unless you have calculated the exact frequencies. Always use proportion or percentage when totals are different.
像“九年级学生中选择蓝色的比例更高”这类说法是合理的。除非你计算出了确切的频数，否则不要说“更多九年级学生喜欢蓝色”。当总数不同时，一定要使用比例或百分比进行比较。
12. Pie Chart Exam Hints and Revision Summary | 饼图考试提示与复习小结
In KS3 Cambridge Maths assessments, you may be asked to read a pie chart, fill in a missing value, or draw one from a table. Always show your working clearly: write the angle or frequency formula, substitute numbers, then give the answer. If a pie chart question mentions ‘estimate’ or ‘give your answer to the nearest…’, round appropriately.
在 KS3 剑桥数学评估中，你可能需要解读饼图、补全缺失值或根据表格绘制饼图。务必清晰地展示你的计算过程：写出角度或频数公式，代入数据，然后给出答案。如果饼图题目中出现“估算”或“给出保留至……的答案”，要进行适当的四舍五入。
| Formula / 公式 | Use / 用途 |
| Angle = (Frequency ÷ Total) × 360° | Calculate sector angle / 计算扇形角 |
| Frequency = (Angle ÷ 360°) × Total | Find frequency from angle / 由角求频数 |
| Percentage = (Angle ÷ 360°) × 100 % | Convert angle to percentage / 角度转百分比 |
| Total = Known Frequency × (360° ÷ Known Angle) | Find total when one sector is known / 知一扇形求总数 |
Keep these formulas in mind, and practise with a protractor. Understanding pie charts will strengthen your data handling skills and prepare you for more advanced topics like histograms and cumulative frequency at IGCSE level.
牢记这些公式，并多多练习用量角器。理解饼图将加强你的数据处理技能，为 IGCSE 阶段的直方图和累积频数等更高级的内容打下基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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