G-7: Angle Properties of Polygons | G-7：多边形的角性质

📚 G-7: Angle Properties of Polygons | G-7：多边形的角性质

Welcome to this revision guide on angle properties of polygons, an essential topic for IGCSE Mathematics. These rules appear frequently in Paper 2 and Paper 4, and mastering them will help you solve a wide range of geometry problems with speed and confidence. In this article, we explore interior angles, exterior angles, regular polygons, and practical problem-solving strategies that follow the IGCSE syllabus closely.

欢迎阅读本篇关于多边形角性质的复习指南，这是 IGCSE 数学中的核心考点，在 Paper 2 和 Paper 4 中频繁出现。掌握这些规律可以帮助你快速且自信地解决各类几何问题。本文将系统讲解内角、外角、正多边形以及贴合 IGCSE 考纲的实用解题策略。

1. The Angle Sum of a Triangle | 三角形的内角和

The sum of the three interior angles inside any triangle always equals 180°. This is one of the most fundamental facts in geometry. It applies to every type of triangle: scalene, isosceles, or equilateral. If the three angles are labelled a, b, and c, then the following relationship always holds.

任何三角形三个内角之和恒等于 180°。这是几何学中最基本的事实之一，适用于所有三角形：不等边三角形、等腰三角形和等边三角形。若三个角分别记为 a、b、c，则以下关系始终成立。

a + b + c = 180°

This fact is the foundation for every polygon angle rule in this article. Whenever you split a larger polygon into triangles, you are using this 180° property as your starting point. In exams, you will often need to combine this rule with parallel-line angle facts or isosceles triangle properties.

这一事实是本文所有多边形角规律的基础。每当你将较大的多边形分割成三角形时，实际上都是在使用这个 180° 的性质作为出发点。在考试中，你常常需要将该规律与平行线角性质或等腰三角形性质结合使用。

2. The Angle Sum of a Quadrilateral | 四边形的内角和

A quadrilateral can be divided into two triangles by drawing one diagonal. Since each triangle has an angle sum of 180°, the total angle sum for a quadrilateral is 2 × 180° = 360°. Therefore, the interior angles of any four-sided polygon always add up to 360°. This is true even for irregular quadrilaterals such as kites and trapeziums.

画一条对角线，可以将一个四边形分成两个三角形。由于每个三角形的内角和为 180°，四边形的内角和为 2 × 180° = 360°。因此，任何四边形的内角之和恒为 360°。即使是不规则的四边形，如风筝形和梯形，这一规律也始终成立。

Sum of interior angles of a quadrilateral = 360° | 四边形内角和 = 360°

This result is extremely useful when one angle of a quadrilateral is unknown. Subtract the three known angles from 360° to find the missing one. Be careful: a square and a rectangle both have four right angles, but a general quadrilateral may have no equal angles at all.

当四边形中有一个角未知时，这个结论非常有用：用 360° 减去三个已知角即可求出未知角。注意：正方形和矩形都有四个直角，但一般四边形的各角可能完全不相等。

3. General Formula for Interior Angles | 内角和的一般公式

For any polygon with n sides, drawing all diagonals from one vertex divides the polygon into (n − 2) triangles. Because each triangle contains 180°, the sum of the interior angles is given by the formula below. This formula works for every polygon, regular or irregular.

对于有 n 条边的多边形，从一个顶点出发画出所有对角线，可以将多边形分成 (n − 2) 个三角形。由于每个三角形内角和为 180°，多边形的内角和可由下面的公式求出。该公式适用于所有多边形，无论是正多边形还是不规则多边形。

S = (n − 2) × 180°

For example, a pentagon (n = 5) has an angle sum of (5 − 2) × 180° = 540°. A hexagon (n = 6) has an angle sum of (6 − 2) × 180° = 720°. The table below lists common polygons and their angle sums, which are well worth memorising for speed in the exam.

例如，五边形（n = 5）的内角和为 (5 − 2) × 180° = 540°。六边形（n = 6）的内角和为 (6 − 2) × 180° = 720°。下表列出了常见多边形及其内角和，这些数值值得熟记，以便在考试中快速作答。

Polygon \| 多边形	Sides n \| 边数 n	Sum of interior angles \| 内角和
Triangle \| 三角形	3	180°
Quadrilateral \| 四边形	4	360°
Pentagon \| 五边形	5	540°
Hexagon \| 六边形	6	720°
Octagon \| 八边形	8	1080°
Decagon \| 十边形	10	1440°

To use the formula in reverse, set S equal to the known angle sum and solve for n. For instance, if a polygon has an angle sum of 1260°, then 1260° = (n − 2) × 180°, giving n − 2 = 7 and n = 9. The polygon is a nonagon.

若要反向使用公式，令 S 等于已知的内角和并解出 n。例如，若一个多边形的内角和为 1260°，则 1260° = (n − 2) × 180°，解得 n − 2 = 7，即 n = 9。该多边形是九边形。

4. Regular Polygons | 正多边形

A regular polygon has all sides equal in length and all interior angles equal in size. Because all n interior angles are identical, one interior angle is simply the total sum divided by n. This gives the important formula shown below.

正多边形的各边长度相等，各内角大小相等。由于 n 个内角完全相同，单个内角就是内角和除以 n。于是得到下面这个重要的公式。

Interior angle of a regular polygon = (n − 2) × 180° ÷ n | 正多边形单个内角 = (n − 2) × 180° ÷ n

For example, an equilateral triangle has (3 − 2) × 180° ÷ 3 = 60° per angle. A square has (4 − 2) × 180° ÷ 4 = 90°. A regular pentagon has 540° ÷ 5 = 108°, and a regular hexagon has 720° ÷ 6 = 120°. These common values appear in many IGCSE questions, so remembering them saves valuable time.

例如，等边三角形的每个角为 (3 − 2) × 180° ÷ 3 = 60°。正方形的每个角为 (4 − 2) × 180° ÷ 4 = 90°。正五边形的每个角为 540° ÷ 5 = 108°，正六边形的每个角为 720° ÷ 6 = 120°。这些常见数值在许多 IGCSE 题目中出现，记住它们能节省宝贵的时间。

It also follows that a regular polygon can never have interior angles of 180° or more, because such a shape would become flat or concave. This simple check can help you spot calculation errors before you finish a problem.

由此可知，正多边形的内角不可能达到 180° 或更大，否则图形会变平或变成凹多边形。这个简单的检查能帮助你在完成题目之前发现计算错误。

5. Exterior Angles | 外角

An exterior angle is formed by extending one side of a polygon beyond a vertex. The most remarkable fact about exterior angles is that their sum is always exactly 360° for any polygon, regardless of how many sides it has. This result holds for irregular polygons as well.

外角是由延长多边形一条边到顶点之外而形成的角。外角最引人注目的性质是：任何多边形（无论有多少条边）的外角和恒等于 360°。这一结论同样适用于不规则多边形。

Sum of exterior angles = 360° | 外角和 = 360°

For a regular polygon, all exterior angles are equal, so each exterior angle is calculated by dividing 360° by the number of sides n. For example, a regular octagon has exterior angles of 360° ÷ 8 = 45°. This formula is the quickest way to find the number of sides when an exterior angle is known.

对于正多边形，所有外角相等，因此每个外角等于 360° 除以边数 n。例如，正八边形的每个外角为 360° ÷ 8 = 45°。当已知外角时，这个公式是求边数最快的方法。

Exterior angle of a regular polygon = 360° ÷ n | 正多边形单个外角 = 360° ÷ n

Notice that as the number of sides increases, the exterior angle decreases. A triangle has exterior angles of 120°, while a dodecagon (n = 12) has exterior angles of only 30°. This inverse relationship is a useful mental check.

注意：随着边数增加，外角逐渐减小。三角形的外角为 120°，而十二边形（n = 12）的外角仅为 30°。这种反比关系是一个有用的心算检查方法。

6. Relationship Between Interior and Exterior Angles | 内角与外角的关系

At any vertex of a polygon, the interior angle and the exterior angle lie on a straight line, so they add up to 180°. This simple relationship connects all the formulas in this topic and allows you to convert between interior and exterior angles instantly.

在多边形的任意一个顶点处，内角与外角位于同一条直线上，因此二者之和为 180°。这个简洁的关系将本主题的所有公式联系起来，使你可以即时在内角与外角之间进行转换。

Interior angle + Exterior angle = 180° | 内角 + 外角 = 180°

For a regular polygon, this gives an alternative way to find the interior angle: first calculate the exterior angle as 360° ÷ n, then subtract from 180°. For example, a regular 12-sided polygon has exterior angle 30° and interior angle 180° − 30° = 150°. Both routes lead to the same answer.

对于正多边形，这提供了求内角的另一种方法：先计算外角为 360° ÷ n，再用 180° 减去该值。例如，正十二边形的外角为 30°，内角为 180° − 30° = 150°。两条路径都会得到相同的答案。

When solving problems, always check which angle you have been given. A common mistake is to use an interior angle in an exterior-angle formula. Reading the question carefully and labelling the diagram is the best defence against this error.

解题时，务必确认题目给出的是内角还是外角。一个常见错误是将内角直接代入外角公式。仔细审题并在图中标注是避免此类错误的最佳方法。

7. Problem-Solving: Finding Unknown Angles | 解题：求未知角

Let us work through a typical exam question. A regular polygon has each exterior angle equal to 24°. How many sides does it have? Since the sum of exterior angles is 360°, we have n = 360° ÷ 24° = 15. The polygon is a regular 15-sided polygon.

让我们解答一道典型考题。一个正多边形的每个外角为 24°，求它的边数。因为外角和为 360°，所以 n = 360° ÷ 24° = 15。该多边形是正十五边形。

Another common question asks for the interior angle of a regular 20-gon. Using the interior angle formula, we get (20 − 2) × 180° ÷ 20 = 3240° ÷ 20 = 162°. As a check, the exterior angle is 360° ÷ 20 = 18°, and 162° + 18° = 180°, which confirms the answer.

另一道常见考题是求正二十边形的一个内角。代入内角公式得 (20 − 2) × 180° ÷ 20 = 3240° ÷ 20 = 162°。检验一下：外角为 360° ÷ 20 = 18°，且 162° + 18° = 180°，验证了答案正确。

Now try a mixed problem. In a quadrilateral, three angles are 90°, 120°, and 85°. Find the fourth angle x. Using the quadrilateral angle sum, x = 360° − 90° − 120° − 85° = 65°. Always write your working clearly, because method marks are awarded in IGCSE examinations.

再尝试一道综合题。在一个四边形中，三个角分别为 90°、120° 和 85°，求第四个角 x。根据四边形内角和，x = 360° − 90° − 120° − 85° = 65°。解题时务必书写清晰，因为 IGCSE 考试会按步骤给分。

For mixed polygons involving parallel lines, combine these rules with alternate angles and corresponding angles. Draw the known points on the diagram and work step by step from the given angles towards the unknown angle, always justifying each line with a written reason.

对于涉及平行线的混合多边形问题，需将这些规则与内错角、同位角结合使用。在图中标出已知角，从给定角逐步推导到未知角，每一步都要写出理由作为依据。

8. Common Exam Questions and Traps | 常见考题与陷阱

The following points summarise the mistakes that students most often make in IGCSE examinations on this topic. Reviewing them carefully can save you from losing easy marks.

以下要点总结了学生在 IGCSE 考试中关于本主题最常见的错误。仔细复习这些内容可以帮助你避免丢失本可轻松获得的分数。

Always check that n represents the number of sides of the polygon, not the number of vertices counted twice. A hexagon has 6 sides, so it is divided into 4 triangles, not 6.

务必确认 n 表示多边形的边数，而不是重复计数的顶点数。六边形有 6 条边，因此被分成 4 个三角形，而不是 6 个。
Do not confuse interior angle with exterior angle. The interior angle is inside the polygon; the exterior angle is outside it on a straight line extension.

不要混淆内角与外角。内角在多边形内部；外角在边的延长线上，位于多边形外部。
Remember that the sum of exterior angles is always 360°, not 180°. The 180° fact applies only to the interior angles of a triangle.

记住外角和恒为 360°，而不是 180°。180° 这一性质仅适用于三角形内角和。
If a polygon is not regular, you may not assume all angles are equal. The formulas for a single interior angle or exterior angle apply only to regular polygons.

如果多边形不是正多边形，不能假设所有角相等。单个内角或外角的公式仅适用于正多边形。
Check that your answers are reasonable. The interior angle of a regular polygon increases as n increases and approaches 180° but never reaches it.

检查答案是否合理。正多边形的内角随着 n 的增大而增大，趋近 180° 但永远不会达到 180°。

9. Practice Problems | 练习题

Use the rules from this article to solve the following practice problems. Attempt each one before reading the answer, and write down your reasons for each step.

请运用本文的规律解答以下练习题。先独立尝试再对照答案，并写出每一步的理由。

Find the sum of the interior angles of a regular dodecagon (12 sides). Answer: (12 − 2) × 180° = 1800°.

求正十二边形的内角和。答案：(12 − 2) × 180° = 1800°。
A regular polygon has an interior angle of 150°. How many sides does it have? Answer: exterior angle = 30°, so n = 360° ÷ 30° = 12 sides.

一个正多边形的内角为 150°，求它的边数。答案：外角 = 30°，所以 n = 360° ÷ 30° = 12 边。
The angles of a pentagon are 100°, 110°, 120°, 130°, and x. Find x. Answer: x = 540° − 460° = 80°.

五边形的五个角分别为 100°、110°、120°、130° 和 x，求 x。答案：x = 540° − 460° = 80°。
Find the number of sides of a polygon whose interior angle sum is 1980°. Answer: (n − 2) × 180° = 1980°, so n − 2 = 11 and n = 13.

已知一个多边形的内角和为 1980°，求边数。答案：(n − 2) × 180° = 1980°，所以 n − 2 = 11，n = 13。

10. Summary | 总结

To succeed in this topic, remember the core results: the interior angles of a triangle sum to 180°; the interior angles of a quadrilateral sum to 360°; a polygon with n sides has interior angle sum (n − 2) × 180°; the exterior angles of any polygon sum to 360°; and each interior angle plus each exterior angle equals 180°. These five facts cover nearly every question in the IGCSE syllabus on polygons.

要在本主题中取得好成绩，请记住以下核心结论：三角形内角和为 180°；四边形内角和为 360°；n 边形内角和为 (n − 2) × 180°；任何多边形的外角和为 360°；每个内角与每个外角之和为 180°。这五个事实几乎涵盖了 IGCSE 考纲中关于多边形的所有题目。

Regular polygon formulas follow directly from these facts: divide the angle sum by n to get one interior angle, and divide 360° by n to get one exterior angle. Practise converting between sides, interior angles, and exterior angles until the process becomes automatic.

正多边形的公式直接源于这些事实：内角和除以 n 得到单个内角，360° 除以 n 得到单个外角。请反复练习在边数、内角和外角之间相互转换，直到整个流程变得熟练自如。

Finally, always draw a clear diagram, label all known angles, and write a reason for every step. This habit not only prevents careless errors but also secures method marks in the examination. Keep practising with past-paper questions, and you will find these angle problems becoming increasingly straightforward.

最后，务必画出清晰的图形，标出所有已知角，并为每一步写出理由。这个习惯不仅能避免粗心错误，还能确保在考试中获得步骤分。坚持用历年真题练习，你会发现这类角度问题变得越来越简单。

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G-7: Angle Properties of Polygons | G-7：多边形的角性质