📚 Interior and Exterior Angles of Polygons: Mastering the 1080° Case | 多边形内角与外角：攻克 1080° 题型
In IGCSE and A-Level mathematics, polygon angle questions may look different on the surface, but they are almost always solved by two simple facts: the interior angle sum formula and the exterior angle sum rule. This article uses the important 1080° case to help you master the whole topic, from deriving the formula to handling exam-style algebra questions.
在 IGCSE 与 A-Level 数学中，多边形角度题表面上看起来各不相同，但几乎都可以用两个简单事实解决：内角和公式与外角和规则。本文以重要的 1080° 题型为核心，从公式推导到考试风格代数题，帮助你完整掌握整个知识点。
1. The Key Idea in One Sentence | 一句话掌握核心
For any convex polygon with n sides, the sum of all interior angles is (n − 2) × 180°.
对于任意有 n 条边的凸多边形，所有内角之和为 (n − 2) × 180°。
Interior angle sum = (n − 2) × 180°
内角和 = (n − 2) × 180°
The exterior angles, taken one at each vertex, always add up to 360°, no matter how many sides the polygon has.
在每个顶点各取一个外角，其总和恒为 360°，无论多边形有多少条边。
Exterior angle sum = 360° (always)
外角和 = 360°（恒定）
n = 8, so it is a perfect example of how one formula unlocks an entire family of questions.
如果你记住这两条规则，你就能高效解决几乎每一道 IGCSE 多边形题目。1080° 题型本质上就是内角和公式中 n = 8 的情形，因此它是“一个公式打通一类题”的最佳范例。
2. Polygon Basics and Notation | 多边形基础与记号
A polygon is a closed flat shape made of straight line segments. Common names include triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9) and decagon (10).
多边形是由直线段组成的封闭平面图形。常见名称包括三角形（3 条边）、四边形（4）、五边形（5）、六边形（6）、七边形（7）、八边形（8）、九边形（9）和十边形（10）。
A regular polygon has all sides equal and all angles equal. An irregular polygon does not. The angle rules work for both regular and irregular polygons, but 'one interior angle' can only be found directly in a regular polygon unless extra information is given.
正多边形所有边相等、所有角相等；不规则多边形则不是。角度规则对正多边形和不规则多边形都适用，但“单个内角”通常只能在正多边形中直接求出，除非题目给出额外信息。
In IGCSE problems, n is usually the number of sides, and a convex polygon means every interior angle is less than 180°. All the formulas in this article assume convex polygons, which is what exam boards such as Cambridge and Edexcel use at this level.
在 IGCSE 题目中，n 通常表示边数，凸多边形指的是每个内角都小于 180°。本文所有公式都假设多边形是凸的，这也是 Cambridge、Edexcel 等考试局在这一阶段的标准设定。
Key notation you should recognise quickly:
你需要快速识别的关键记号：
- n = number of sides = number of vertices = number of interior angles；n = 边数 = 顶点数 = 内角个数
- Interior angle = the angle inside the polygon at a vertex；内角 = 顶点处多边形内部的角
- Exterior angle = the angle between one side extended and the next side；外角 = 一边延长线与相邻边之间的角
- Regular = all sides and all angles equal；正多边形 = 各边相等、各内角相等
3. Deriving the Interior Angle Sum Formula | 推导内角和公式
Understanding the derivation makes the formula almost impossible to forget and shows up in 'explain' or 'show that' exam questions.
不要只背 (n − 2) × 180°。理解推导过程能让这个公式几乎不可能忘记，而且它在“解释”或“证明”类考题中会直接出现。
Choose one vertex of the polygon and draw all the diagonals from it to the other non-adjacent vertices. These diagonals split the polygon into triangles.
选取多边形的一个顶点，从它向所有不相邻的顶点作对角线。这些对角线把多边形分割成若干个三角形。
gonals from one vertex. Since each triangle has an interior angle sum of 180°, the polygon's interior angle sum must be (n − 2) × 180°.
任何 n 边形从同一个顶点引对角线，都会被恰好分成 (n − 2) 个三角形。由于每个三角形内角和为 180°，多边形的内角和必然是 (n − 2) × 180°。
Triangles formed = n − 2 → Sum = (n − 2) × 180°
分出的三角形数 = n − 2 → 内角和 = (n − 2) × 180°
quadrilateral, (4 − 2) × 180° = 360°; for a pentagon, (5 − 2) × 180° = 540°. All correct.
用熟悉的图形快速检验：三角形 (3 − 2) × 180° = 180°；四边形 (4 − 2) × 180° = 360°；五边形 (5 − 2) × 180° = 540°。全部正确。
4. Solving the 1080° Case: Finding the Number of Sides | 攻克 1080° 题型：求边数
The classic 1080° question gives you the interior angle sum and asks for the number of sides, the name of the polygon, or the size of one interior angle. Let us work through it step by step.
经典的 1080° 题目会给出内角和，要求你求边数、说出多边形名称，或求某个内角的大小。我们一步一步来做。
Step 1: Write down the formula.
第一步：写出公式。
(n − 2) × 180° = 1080°
Step 2: Divide both sides by 180°.
第二步：两边同时除以 180°。
n − 2 = 1080° ÷ 180° = 6
Step 3: Add 2 to both sides.
第三步：两边同时加 2。
n = 6 + 2 = 8
So a polygon whose interior angles add up to 1080° has 8 sides. Its name is an octagon.
因此，内角和为 1080° 的多边形有 8 条边，名称是八边形。
which is a useful signal: if the given sum is not divisible by 180°, then no such polygon exists, and the correct answer is 'impossible'.
1080° ÷ 180° = 6 是整除，这是一个有用的信号：如果给出的内角和不能被 180° 整除，那么这样的多边形不存在，正确答案就是“不可能”。
5. One Interior Angle of a Regular Octagon | 正八边形的每个内角
now n = 8, finding a single interior angle in a regular octagon is a one-line calculation.
一旦确定 n = 8，求正八边形单个内角就是一行计算的事。
One interior angle = (n − 2) × 180° ÷ n
每个内角 = (n − 2) × 180° ÷ n
tuting n = 8 gives 1080° ÷ 8 = 135°. So every interior angle of a regular octagon is 135°.
代入 n = 8 得 1080° ÷ 8 = 135°。因此正八边形的每个内角都是 135°。
This 135° value appears constantly in exam papers, especially in combination with tiling questions, because 135° does not divide 360° exactly, so regular octagons alone cannot tile a plane.
这个 135° 在试卷中反复出现，尤其在密铺（tiling）题中，因为 135° 不能整除 360°，所以正八边形单独无法密铺平面。
Be careful with the wording: 'each interior angle is 135°' is only true for a regular octagon. An irregular octagon also has an interior angle sum of 1080°, but its individual angles can be anything that adds to 1080°.
注意表述差异：只有正八边形才能说“每个内角都是 135°”。不规则八边形的内角和同样是 1080°，但它的各个内角只要加起来等于 1080° 就可以是任意值。
6. The Exterior Angle Sum Rule: Always 360° | 外角和恒为 360°
The exterior angle sum rule is the most powerful shortcut in this topic, because it does not depend on n at all.
外角和规则是本知识点中最强大的捷径，因为它完全与 n 无关。
Imagine walking around the polygon along its edges. At each vertex you must turn through the exterior angle. By the time you return to your starting point facing the original direction, you have completed exactly one full turn, which is 360°.
想象你沿着多边形的边行走。在每个顶点你都要转过一个外角。当你回到起点并朝向原来的方向时，你恰好完成了一整圈，也就是 360°。
Sum of exterior angles (one per vertex) = 360°
外角和（每顶点取一个）= 360°
For a regular polygon, each exterior angle is equal, so:
对于正多边形，每个外角相等，所以：
One exterior angle = 360° ÷ n
每个外角 = 360° ÷ n
terior angle is 360° ÷ 8 = 45°. Notice how much faster this is than subtracting from 180°.
对于正八边形，每个外角是 360° ÷ 8 = 45°。注意这比用 180° 相减要快得多。
7. The Link 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 are supplementary.
在多边形的任一顶点处，内角与外角位于同一条直线上，因此它们互补。
Interior angle + Exterior angle = 180°
内角 + 外角 = 180°
180°. Correct. This relationship lets you switch between the two rules whenever it is quicker.
用八边形检验：135° + 45° = 180°。正确。这个关系让你可以在两条规则之间自由切换，选择更快的路径。
or angles is n × 180°. Subtracting the exterior sum of 360° gives n × 180° − 360° = (n − 2) × 180°, which matches the formula derived earlier.
这个关系还给出了内角和的第二种独立证明。若有 n 个顶点，则所有内角与所有外角之和为 n × 180°。减去外角和 360°，得 n × 180° − 360° = (n − 2) × 180°，与前面推导的公式一致。
Exam tip: when a question mentions 'exterior angle', always check whether it means the true exterior angle (180° − interior) or the angle turned through while walking. For convex polygons these are the same value.
考试提示：题目提到“外角”时，先确认它指的是真正的外角（180° − 内角），还是行走时转过的角。对于凸多边形，这两个值是相同的。
8. Worked Backwards: Finding n from One Angle | 反向求解：由一个角求边数
Exam questions often give you one interior or exterior angle of a regular polygon and ask for the number of sides. The exterior angle route is almost always faster.
考题常常给出正多边形的一个内角或外角，要求你求边数。用外角路线几乎总是更快。
Example: each interior angle of a regular polygon is 156°. Find the number of sides.
例：正多边形的每个内角为 156°，求边数。
Method A (exterior angle): exterior angle = 180° − 156° = 24°, so n = 360° ÷ 24° = 15.
方法 A（外角法）：外角 = 180° − 156° = 24°，所以 n = 360° ÷ 24° = 15。
= 156° × n, so 180n − 360 = 156n, giving 24n = 360 and n = 15. Same answer, more algebra.
方法 B（内角和法）：(n − 2) × 180° = 156° × n，即 180n − 360 = 156n，得 24n = 360，n = 15。答案相同，但代数步骤更多。
. Then n = 360° ÷ 30° = 12, so it is a regular dodecagon, and each interior angle is 150°.
例：正多边形的每个外角为 30°。则 n = 360° ÷ 30° = 12，是正十二边形，每个内角为 150°。
ior angle less than 180°, and each exterior angle less than 120° (since n = 3 gives 120°).
务必做合理性检验：正多边形的 n ≥ 3，每个内角小于 180°，每个外角小于 120°（因为 n = 3 时外角为 120°）。
9. Irregular Polygons and Algebra-Style Questions | 不规则多边形与代数式题型
In irregular polygon questions, you cannot use 360° ÷ n or 1080° ÷ n, because the angles are not equal. Instead you form an equation from the interior angle sum.
在不规则多边形题目中，不能使用 360° ÷ n 或 1080° ÷ n，因为各个角不相等。此时应利用内角和列出方程。
Example: the interior angles of a hexagon are x, x + 10, x + 20, x + 30, x + 40 and x + 50 degrees. Find x.
例：某六边形的内角依次为 x、x + 10、x + 20、x + 30、x + 40、x + 50（度）。求 x。
720°. The six expressions add to 6x + 150. So 6x + 150 = 720, giving 6x = 570 and x = 95.
六边形内角和为 (6 − 2) × 180° = 720°。六个表达式之和为 6x + 150。于是 6x + 150 = 720，得 6x = 570，x = 95。
you would first identify n = 8, then set the sum of the eight expressions equal to 1080°.
如果同一题改为内角和为 1080°，你应先确定 n = 8，再把八个表达式之和设为 1080°。
Another common style: a polygon has n sides and the sum of its interior angles is between two given values. Find the possible values of n. Here you use inequalities rather than exact division.
另一种常见题型：某多边形有 n 条边，其内角和介于两个给定值之间。求 n 的所有可能值。此时要使用不等式，而不是精确整除。
10. Common Traps and Exam Mistakes | 常见陷阱与易错点
Trap 1: using 360° ÷ n when the question asks for an interior angle. Remember 360° ÷ n gives the exterior angle; subtract from 180° if needed.
陷阱 1：题目要求内角时却用了 360° ÷ n。记住 360° ÷ n 得到的是外角；如需内角，再用 180° 相减。
Trap 2: assuming a polygon is regular when it is not. If the question does not say 'regular', you may only use the sums, not the equal-division shortcuts.
陷阱 2：在题目未说明正多边形时擅自按正多边形处理。如果没有写“regular（正）”，就只能使用和，而不能用等分捷径。
is 360° regardless of n. Students often try to compute (n − 2) × 180° for exterior angles.
陷阱 3：忘记外角和恒为 360°，与 n 无关。很多学生错误地用 (n − 2) × 180° 去算外角。
Trap 4: mixing up the polygon names. Eight sides is an octagon, not a hexagon; the 1080° sum belongs to the octagon.
陷阱 4：混淆多边形名称。8 条边是八边形（octagon），不是六边形；1080° 对应的正是八边形。
Trap 5: losing the degree symbol or the units in the final answer. Angle answers must carry the ° symbol.
陷阱 5：最终答案漏写度数符号。角度的答案必须带 ° 符号。
Trap 6: in 'show that' questions, jumping straight to the answer without showing the substitution. Always write the formula, then substitute, then solve.
陷阱 6：在“证明”类题目中直接跳到答案。务必写出公式、代入、求解三步。
11. Exam-Style Worked Examples | 考试风格例题精讲
Example 1: The sum of the interior angles of a polygon is 1080°. Find the number of sides and the size of each exterior angle if the polygon is regular.
例 1：某多边形的内角和为 1080°。求边数；若该多边形为正多边形，求每个外角的大小。
° = 1080° → n − 2 = 6 → n = 8. Each exterior angle = 360° ÷ 8 = 45°. Answer: 8 sides, 45°.
解：(n − 2) × 180° = 1080° → n − 2 = 6 → n = 8。每个外角 = 360° ÷ 8 = 45°。答案：8 条边，45°。
Example 2: A regular polygon has an interior angle of 135°. Show that it has 8 sides.
例 2：某正多边形的内角为 135°。证明它有 8 条边。
Solution: exterior angle = 180° − 135° = 45°. Then n = 360° ÷ 45° = 8, as required.
解：外角 = 180° − 135° = 45°。于是 n = 360° ÷ 45° = 8，证毕。
Example 3: Three of the interior angles of a pentagon are 100°, 110° and 120°. The remaining two angles are equal. Find them.
例 3：某五边形有三个内角为 100°、110° 和 120°。其余两个内角相等。求这两个角。
540°. Remaining total = 540° − (100° + 110° + 120°) = 210°. Each angle = 210° ÷ 2 = 105°.
解：五边形内角和 = (5 − 2) × 180° = 540°。剩余总和 = 540° − (100° + 110° + 120°) = 210°。每个角 = 210° ÷ 2 = 105°。
Example 4: Explain why a regular polygon cannot have an interior angle of 1080°.
例 4：解释为什么正多边形的内角不可能是 1080°。
Solution: an interior angle of a convex polygon must be less than 180°. Since 1080° > 180°, it cannot be an interior angle; 1080° is an interior angle sum, not a single angle.
解：凸多边形的内角必须小于 180°。由于 1080° > 180°，它不可能是内角；1080° 是内角和，而不是单个角。
12. Quick Reference Table and Checklist | 速查表与自查清单
Use this table to memorise the most common interior angle sums. The 1080° row is highlighted as the focus of this article.
用下表记住最常见的内角和。1080° 一行即是本文的重点。
| Polygon 多边形 | n | Interior sum 内角和 | Each angle (regular) 正多边形每角 | Each exterior 每个外角 |
| Triangle 三角形 | 3 | 180° | 60° | 120° |
| Quadrilateral 四边形 | 4 | 360° | 90° | 90° |
| Pentagon 五边形 | 5 | 540° | 108° | 72° |
| Hexagon 六边形 | 6 | 720° | 120° | 60° |
| Heptagon 七边形 | 7 | 900° | ≈128.57° | ≈51.43° |
| Octagon 八边形 | 8 | 1080° | 135° | 45° |
| Nonagon 九边形 | 9 | 1260° | 140° | 40° |
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