📚 Shape and Space 9 | 图形与空间 9
Welcome to this comprehensive revision guide for the Edexcel IGCSE Mathematics unit Shape and Space 9. This section brings together the most testable ideas about 2D and 3D measurement, circle geometry, trigonometry, transformations and vectors. We will walk through each topic in a clear, step-by-step manner, with the exact formulas and techniques you need for the exam.
欢迎阅读 Edexcel IGCSE 数学 “图形与空间 9” 单元的全面复习指南。本单元汇集了二维与三维度量、圆几何、三角学、变换与向量中最常考查的核心内容。我们将以清晰、循序渐进的方式逐一讲解每个主题，并提供考试中所需的全部公式与解题技巧。
1. Perimeter and Area of 2D Shapes | 二维图形的周长与面积
The perimeter is the total distance around the outside of a shape. For a rectangle, the perimeter is 2 × (length + width), while the area is length × width. For a triangle, the area is ½ × base × height, which works for any triangle as long as the height is measured perpendicular to the base.
周长是图形外边界一周的总长度。对于矩形，周长等于 2 × (长 + 宽)，面积等于长 × 宽。对于三角形，面积等于 ½ × 底 × 高，只要高垂直于底边，该公式适用于任何三角形。
For a parallelogram, the area is base × perpendicular height. For a trapezium, the area is ½ × (a + b) × h, where a and b are the two parallel sides and h is the perpendicular distance between them.
平行四边形的面积等于底 × 高。梯形的面积等于 ½ × (上底 + 下底) × 高，其中 a 和 b 是两条平行边，h 是它们之间的垂直距离。
| Shape | Perimeter | Area |
| Rectangle | 2(l + w) | l × w |
| Triangle | sum of three sides | ½ × b × h |
| Parallelogram | 2(a + b) | b × h |
| Trapezium | sum of four sides | ½ × (a + b) × h |
2. Circles: Circumference and Area | 圆：周长与面积
The circumference of a circle is the distance around it. It can be calculated using C = π × d or C = 2 × π × r, where d is the diameter and r is the radius. The area of a circle is A = π × r². In exam questions, always check whether you are given the radius or the diameter before substituting values.
圆的周长是绕圆一周的距离，可用公式 C = π × d 或 C = 2 × π × r 计算，其中 d 是直径，r 是半径。圆的面积公式为 A = π × r²。在考试中，代入数值前务必先确认题目给的是半径还是直径。
An arc is part of the circumference, and a sector is part of the area. For a sector with angle θ (in degrees), the arc length is (θ/360) × 2πr and the sector area is (θ/360) × πr². Remember to leave your answer in terms of π when the question asks for an exact value.
弧是圆周的一部分，扇形是圆面积的一部分。对于圆心角为 θ（以度为单位）的扇形，弧长等于 (θ/360) × 2πr，扇形面积等于 (θ/360) × πr²。当题目要求精确值时，请将答案保留为含 π 的形式。
Arc length = (θ/360) × 2πr, Sector area = (θ/360) × πr²
3. Surface Area and Volume of 3D Shapes | 三维图形的表面积与体积
Prisms are 3D shapes with a uniform cross-section. The volume of any prism is given by V = area of cross-section × length. For a cuboid, this becomes length × width × height. The surface area is the sum of the areas of all faces.
棱柱是具有统一横截面的三维图形。任何棱柱的体积公式为 V = 横截面积 × 长度。对于长方体，它变为 长 × 宽 × 高。表面积是所有面的面积之和。
For a cylinder, the volume is V = πr²h and the curved surface area is 2πrh. The total surface area of a closed cylinder is 2πr² + 2πrh. For a sphere, the volume is (4/3)πr³ and the surface area is 4πr². For a cone, the volume is (1/3)πr²h and the curved surface area is πrl, where l is the slant height.
圆柱的体积公式为 V = πr²h，侧面积为 2πrh。封闭圆柱的总表面积为 2πr² + 2πrh。球的体积为 (4/3)πr³，表面积为 4πr²。圆锥的体积为 (1/3)πr²h，侧面积为 πrl，其中 l 是母线长。
4. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies to right-angled triangles. If a and b are the two shorter sides and c is the hypotenuse, then a² + b² = c². To find the hypotenuse, take the square root of the sum of the squares of the other two sides.
勾股定理适用于直角三角形。若 a 和 b 是两条直角边，c 是斜边，则 a² + b² = c²。求斜边时，将另外两边的平方相加后再开平方根即可。
To find one of the shorter sides, rearrange the formula: a² = c² − b², then take the square root. In exam questions, look for right angles in diagrams of cuboids, cones or compound shapes. Always identify the hypotenuse first: it is the side opposite the right angle.
求一条直角边时，可将公式变形为 a² = c² − b²，然后开平方。在考试中，注意在长方体、圆锥或复合图形中寻找直角。务必先辨认斜边：它是直角所对的边。
a² + b² = c², c = √(a² + b²), a = √(c² − b²)
5. Trigonometry in Right-Angled Triangles | 直角三角形中的三角学
Trigonometry links the angles and sides of a right-angled triangle. The three ratios are sine, cosine and tangent, commonly remembered as SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
三角学将直角三角形的角与边联系起来。三个基本比值为正弦、余弦和正切，常以 SOHCAHTOA 口诀记忆：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
When finding a missing side, choose the correct ratio and solve the equation. When finding a missing angle, use the inverse functions: sin⁻¹, cos⁻¹ or tan⁻¹. In higher-level papers, you may also apply the sine rule and cosine rule to non-right-angled triangles, but the foundation is always SOHCAHTOA.
求缺失边时，选择正确的比值并解方程。求缺失角时，使用反函数：sin⁻¹、cos⁻¹ 或 tan⁻¹。在较高难度的试卷中，还可能需要将正弦定理和余弦定理应用于非直角三角形，但基础始终是 SOHCAHTOA。
6. Bearings and Angle Rules | 方位角与角度法则
A bearing is an angle measured clockwise from north, always written as three digits. For example, a bearing of 047° means 47 degrees clockwise from north. To find a bearing from point B to point A when you know the bearing from A to B, add or subtract 180°.
方位角是从正北方向顺时针测量的角度，始终用三位数表示。例如，方位角 047° 表示从正北顺时针旋转 47 度。当已知从 A 到 B 的方位角时，求从 B 到 A 的方位角需要加或减 180°。
You must also be fluent in basic angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal, and alternate and corresponding angles formed by parallel lines are equal or supplementary.
你还必须熟练掌握基本角度事实：同一直线上的角之和为 180°，围绕一点的角之和为 360°，对顶角相等，平行线形成的同位角和内错角相等或互补。
7. Loci and Construction | 轨迹与尺规作图
A locus is a set of points that satisfy a given condition. Common loci include the set of points at a fixed distance from a point (a circle) and the set of points equidistant from two fixed points (the perpendicular bisector of the segment joining them).
轨迹是满足给定条件的点的集合。常见的轨迹包括：与一个定点距离固定的点的集合（即圆），以及与两个定点距离相等的点的集合（即连接两点的线段的垂直平分线）。
In construction questions, use a compass and ruler carefully. To draw the locus of points at a fixed distance from a line, draw two semicircles at the ends and two parallel lines. Shade regions when the question asks for a set of points satisfying multiple conditions, such as being within 3 cm of a point and closer to one line than another.
在尺规作图题中，请小心使用圆规和直尺。要画出与一条直线距离固定的点的轨迹，需在两端画两个半圆并画两条平行线。当题目要求满足多个条件的点集时，需要涂阴影区域，例如距离某点 3 cm 以内且更靠近某一条直线。
8. Transformations | 几何变换
Transformations move or change a shape. There are four main types: reflection, rotation, translation and enlargement. A reflection produces a mirror image across a mirror line; the perpendicular distance from each point to the line is preserved.
变换会使图形移动或改变。共有四种主要类型：反射、旋转、平移和放大。反射以一条镜面直线产生镜像；每个点到该直线的垂直距离保持不变。
A rotation turns a shape about a fixed centre through a given angle, usually clockwise or anticlockwise. A translation slides a shape by a vector, and an enlargement changes the size by a scale factor, with the position determined by a centre of enlargement.
旋转使图形围绕固定中心旋转给定角度，通常为顺时针或逆时针。平移通过向量滑动图形。放大以缩放因子改变图形大小，其位置由放大中心决定。
When describing a transformation, always include all details: for reflection, state the mirror line; for rotation, state the centre, angle and direction; for translation, state the vector; for enlargement, state the scale factor and centre. For negative scale factors, the image appears on the opposite side of the centre and inverted.
在描述变换时，务必给出全部细节：反射需说明镜面直线；旋转需说明中心、角度和方向；平移需说明向量；放大需说明缩放因子和中心。当缩放因子为负数时，图形出现在中心的另一侧且被倒置。
9. Vectors | 向量
A vector has both magnitude and direction. It is written as a column vector, such as (3, 2) meaning 3 units right and 2 units up. To add vectors, add the corresponding components. To multiply a vector by a scalar, multiply each component by the scalar.
向量同时具有大小和方向。它常写成列向量形式，例如 (3, 2) 表示向右 3 个单位、向上 2 个单位。向量相加时，将对应分量相加。向量乘以标量时，将每个分量乘以该标量。
If point M is the midpoint of AB, then the position vector of M is (a + b)/2. To find a vector from A to B, subtract the position vector of A from that of B: AB = b − a. Two vectors are equal if they have the same magnitude and direction; they are parallel if one is a scalar multiple of the other.
若 M 是 AB 的中点，则 M 的位置向量为 (a + b)/2。要求从 A 到 B 的向量，用 B 的位置向量减去 A 的位置向量：AB = b − a。两个向量大小和方向都相同则相等；一个向量是另一个的标量倍数则它们平行。
10. Compound Shapes and Problem-Solving Strategy | 复合图形与解题策略
Many exam questions combine several topics from this unit. A compound shape may require you to split an irregular region into rectangles, triangles and sectors, then add or subtract their areas. Always draw a clear diagram and label known lengths before starting calculations.
许多考试题目会综合本单元的多个知识点。复合图形可能需要你将不规则区域分割为矩形、三角形和扇形，再相加或相减其面积。开始计算前，务必画出清晰的示意图并标注已知长度。
A strong problem-solving routine is: read the question twice, identify the relevant formula, substitute carefully, solve step-by-step, and check that the units are consistent. Round only at the final step unless the question gives a specific degree of accuracy. Reviewing past paper questions on circle theorems and 3D trigonometry is especially valuable for reaching the highest grades.
一个有效的解题流程是：读题两遍，识别相关公式，仔细代入，逐步求解，并检查单位是否一致。除非题目指定精确度，否则只在最后一步进行四舍五入。复习历年真题中关于圆定理和三维三角学的题目，对冲刺高分尤其有价值。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply