📚 IGCSE Mathematics: Geometry and Trigonometry Essentials | IGCSE 数学：几何与三角学核心要点
Geometry and trigonometry form a large part of the IGCSE Mathematics syllabus, linking visual reasoning with algebraic calculation. This article reviews the core facts, formulas and problem-solving habits you need for both Core and Extended papers.
几何与三角学在 IGCSE 数学大纲中占据很大比重，它将图形推理与代数计算紧密结合。本文回顾核心事实、公式以及 Core 和 Extended 考试都需要的解题习惯。
1. Angles and parallel lines | 角与平行线
When two parallel lines are cut by a transversal, three angle relationships are essential: corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°.
当两条平行线被一条横截线所截时，有三种关键角度关系：同位角相等、内错角相等、同旁内角之和为 180°。
Always identify the parallel lines first, then mark the transversal. If the diagram does not look parallel, do not assume angle facts unless the question states or shows parallel lines.
解题时先识别平行线，再标出横截线。如果图中没有明确给出平行线，不要随意假设角度关系，必须依据题目条件。
- Corresponding angles: equal | 同位角：相等
- Alternate angles: equal | 内错角：相等
- Co-interior angles: sum to 180° | 同旁内角：相加为 180°
co-interior angles: x + y = 180°
2. Polygons and interior/exterior angles | 多边形与内外角
on is given by (n - 2) × 180°. This comes from splitting the polygon into n - 2 triangles.
n 边形内角和公式为 (n - 2) × 180°。这是因为多边形可以分成 n - 2 个三角形。
The sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides. For a regular polygon, each exterior angle is 360° ÷ n and each interior angle is 180° - exterior angle.
任何凸多边形的外角和始终为 360°，与边数无关。对于正多边形，每个外角为 360° ÷ n，每个内角等于 180° - 外角。
Interior angle sum = (n - 2) × 180°
Use exterior angles when the question mentions a regular polygon or asks for the number of sides; it is often faster than working with interior angles.
遇到正多边形或求边数的问题时，优先使用外角，通常比用内角计算更快。
3. Pythagoras' theorem | 毕达哥拉斯定理
Pythagoras' triangles. If the hypotenuse is c and the other two sides are a and b, then a² + b² = c².
毕达哥拉斯定理只适用于直角三角形。若斜边为 c，另外两条直角边为 a 和 b，则 a² + b² = c²。
- b². Always take the positive square root at the end because lengths cannot be negative.
求直角边时需要变形：a² = c² - b²。最后要取正的平方根，因为长度不能为负。
In 3D problems, apply Pythagoras twice or use the extension d² = x² + y² + z² for a cuboid's space diagonal.
在三维问题中，可以两次使用毕达哥拉斯定理，或对长方体的空间对角线使用 d² = x² + y² + z²。
a² + b² = c²
4. Trigonometric ratios | 三角比
: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.
对于直角三角形，三个基本三角比由 SOH CAH TOA 定义：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Label the sides relative to the angle you are using, not the right angle. Then choose the ratio that connects the known side and the side you want to find.
解题时先根据所给角标出对边、邻边和斜边，注意是相对于该角而不是直角。然后选择连接已知边和未知边的三角比。
Know exact values for 30°, 45° and 60°: sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, and so on. These appear without a calculator on many papers.
需要熟记 30°、45° 和 60° 的精确值：sin 30° = 1/2、cos 30° = √3/2、tan 45° = 1 等。这些常出现在不允许使用计算器的题目中。
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
5. Sine and cosine rules | 正弦与余弦定理
in C. Use it when you know two angles and one side, or two sides and a non-included angle.
对于非直角三角形，正弦定理将边与对角联系起来：a/sin A = b/sin B = c/sin C。当已知两角一边，或两边及一个非夹角时使用。
n you know two sides and the included angle, or all three sides: a² = b² + c² - 2bc cos A.
余弦定理用于已知两边及其夹角，或已知三边的情况：a² = b² + c² - 2bc cos A。
(180° - x). Check whether the angle should be acute or obtuse from the diagram or context.
用正弦定理求角时要注意可能存在的多解情况，因为 sin x = sin(180° - x)。应根据图形或题意判断角是锐角还是钝角。
Sine rule: a/sin A = b/sin B = c/sin C
Cosine rule: a² = b² + c² - 2bc cos A
6. Area of a triangle using trigonometry | 用三角学求三角形面积
= ½ ab sin C, where a and b are the lengths of two sides and C is the angle between them.
任意三角形的面积都可以用两边及其夹角求得：面积 = ½ ab sin C，其中 a 和 b 是两边长度，C 是它们的夹角。
This formula is especially useful when the perpendicular height is not given or is difficult to calculate. It is also used in the proof of the sine rule.
当题目没有给出垂直高度或高度难以计算时，这个公式尤其有用。它也是正弦定理证明的基础。
Remember to use the included angle, not any other angle. In mixed questions, you may need the cosine rule first to find an angle, then the area formula.
一定要使用两边之间的夹角，而不是其他角。在综合题中，可能需要先用余弦定理求出角度，再代入面积公式。
Area = ½ ab sin C
7. Circles: circumference and area | 圆的周长与面积
hese formulas are given in the exam but must be applied correctly with diameter and radius.
对于半径为 r 的圆，周长为 C = 2πr，面积为 A = πr²。考试会提供这些公式，但必须正确区分直径和半径。
in degrees, the arc length is l = (θ/360) × 2πr and the sector area is A = (θ/360) × πr².
对于圆心角为 θ（以度为单位）的扇形，弧长为 l = (θ/360) × 2πr，扇形面积为 A = (θ/360) × πr²。
Be careful with units: if the radius is in cm, the circumference is in cm and the area is in cm². If the question gives the diameter, divide it by 2 first.
注意单位：如果半径以 cm 为单位，周长为 cm，面积为 cm²。如果题目给的是直径，要先除以 2 得到半径。
Circumference = 2πr, Area = πr²
8. Volume and surface area of 3D shapes | 三维图形的体积与表面积
al area × length, cylinder volume = πr²h, sphere volume = 4/3 πr³, cone volume = 1/3 πr²h.
常见公式包括：长方体体积 = 长 × 宽 × 高，棱柱体积 = 横截面积 × 长，圆柱体积 = πr²h，球体积 = 4/3 πr³，圆锥体积 = 1/3 πr²h。
Surface area is the total area of all faces. For a cylinder, it is 2πr² + 2πrh. For a cone, it is πr² + πrl, where l is the slant height.
表面积是所有面的总面积。圆柱的表面积为 2πr² + 2πrh。圆锥的表面积为 πr² + πrl，其中 l 是斜高。
For composite solids, break the shape into standard parts, calculate separately, then add or subtract. Check whether the question asks for curved surface area or total surface area.
对于组合体，应拆分成标准几何体分别计算，然后相加或相减。注意题目要求的是侧面积还是总表面积。
Prism volume = cross-sectional area × length
9. Congruence and similarity | 全等与相似
Congruent shapes are identical in shape and size. The four congruence tests for triangles are SSS, SAS, ASA and RHS (right angle, hypotenuse, side).
全等图形形状和大小完全相同。三角形全等的四种判定方法是 SSS、SAS、ASA 和 RHS（直角、斜边、一边）。
Similar shapes have the same shape but not necessarily the same size. Their corresponding angles are equal, and corresponding sides are in the same ratio.
相似图形形状相同但大小不一定相同。对应角相等，对应边成相同比例。
If the length scale factor is k, then the area scale factor is k² and the volume scale factor is k³. This is a very common exam idea in both 2D and 3D problems.
如果长度比例因子为 k，则面积比例因子为 k²，体积比例因子为 k³。这是二维和三维问题中非常常见的考点。
Length scale factor = k, Area scale factor = k², Volume scale factor = k³
10. Exam tips and common mistakes | 考试技巧与常见错误
Always write the formula first, substitute carefully, and show your working. Even if the final answer is wrong, method marks can be awarded.
做题时先写公式，再仔细代入，并展示计算过程。即使最终答案错误，也可能获得方法分。
Check your calculator mode: it should be in degrees for IGCSE geometry unless the question uses radians. Round answers only at the end, not after each step.
检查计算器模式：除非题目使用弧度，IGCSE 几何中角度通常用度。只在最后一步舍入答案，不要在中间步骤四舍五入。
Common mistakes include mixing up sine and cosine rules, using the wrong angle in the area formula, forgetting that exterior angles sum to 360°, and giving area answers without squared units.
常见错误包括混淆正弦定理和余弦定理、面积公式中用错角、忘记外角和为 360°、面积答案缺少平方单位。
Draw clear diagrams when a question has no diagram. Label unknown sides with variables and mark right angles, parallel lines and equal sides as you read.
如果题目没有图形，应自己画出清晰示意图。用变量标出未知边，并在读题时标出直角、平行线和相等的边。
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