📚 Shape and Space 7 | 形状与空间 7
Welcome to this revision guide on Shape and Space, part of the Edexcel IGCSE Mathematics syllabus. Here we will recap the key facts, formulas and problem-solving techniques you need for your exams.
欢迎阅读这份关于“形状与空间”的复习指南，它对应 Edexcel IGCSE 数学考纲。我们将在这里回顾考试所需的关键性质、公式和解题技巧。
1. Angle Facts and Parallel Lines | 角度基本性质与平行线
Key angle facts: Angles on a straight line add up to 180°, angles around a point add up to 360°, and vertically opposite angles are equal.
关键角度性质：平角上的角度之和为 180°，周角上的角度之和为 360°，对顶角相等。
- Alternate angles (Z pattern): Equal to each other when a transversal crosses parallel lines.
- Alternate angles (Z pattern): 两条平行线被一条截线所截时，内错角（Z 形）相等。
- Corresponding angles (F pattern): Equal to each other.
- Corresponding angles (F pattern): 同位角（F 形）相等。
- Allied angles (C or U pattern): Sum to 180°.
- Allied angles (C or U pattern): 同旁内角（C/U 形）之和为 180°。
a = b (alternate), c = d (corresponding), e + f = 180° (allied)
a = b（内错角），c = d（同位角），e + f = 180°（同旁内角）
When solving angle problems, always state the reason using the correct geometric vocabulary.
在解角度问题时，务必使用准确的几何术语来说明理由。
2. Triangles and Quadrilaterals | 三角形与四边形
A triangle has interior angles summing to 180°. Triangles are classified by their sides or angles.
三角形内角和为 180°。三角形按边或角分类。
- Equilateral: All sides equal, all angles 60°.
- Equilateral / 等边三角形：三边相等，每个角都是 60°。
- Isosceles: Two sides equal, base angles equal.
- Isosceles / 等腰三角形：两腰相等，底角相等。
- Scalene: All sides and all angles different.
- Scalene / 不等边三角形：三边和三角均不相等。
- Right-angled: Contains one 90° angle.
- Right-angled / 直角三角形：含有一个 90° 角。
Quadrilaterals have interior angles summing to 360°. Common quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapeziums and kites.
四边形内角和为 360°。常见四边形包括正方形、长方形、平行四边形、菱形、梯形和筝形。
| Shape / 图形 | Key properties / 关键性质 |
| Square 正方形 | All sides equal, all angles 90°, diagonals equal and perpendicular 四边相等，四角90°，对角线相等且垂直 |
| Rectangle 长方形 | Opposite sides equal, all angles 90°, diagonals equal 对边相等，四角90°，对角线相等 |
| Parallelogram 平行四边形 | Opposite sides parallel and equal, opposite angles equal 对边平行且相等，对角相等 |
| Rhombus 菱形 | All sides equal, diagonals bisect at 90° 四边相等，对角线互相垂直平分 |
| Trapezium 梯形 | One pair of opposite sides parallel 一对对边平行 |
3. Angles in Polygons | 多边形内角与外角
For a polygon with n sides, the sum of interior angles is given by the formula below.
对于有 n 条边的多边形，其内角和由下面的公式给出。
Sum of interior angles = (n − 2) × 180°
内角和 = (n − 2) × 180°
The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle is 360° ÷ n.
任何多边形的外角和均为 360°。正多边形中，每个外角等于 360° ÷ n。
Interior angle + Exterior angle = 180° (regular polygon)
内角 + 外角 = 180°（正多边形）
ample: A regular hexagon has n = 6. Interior angle = (6 − 2) × 180° ÷ 6 = 720° ÷ 6 = 120°.
例如：正六边形 n = 6。内角 = (6 − 2) × 180° ÷ 6 = 720° ÷ 6 = 120°。
4. Circle Properties | 圆的性质
You must remember the following key facts about circles.
你必须牢记以下关于圆的重要事实。
- The angle at the centre is twice the angle at the circumference when both subtend the same arc.
- 圆心角是圆周角的两倍，当两者对应同一条弧时。
- The angle in a semicircle is 90°.
- 半圆内的圆周角是 90°。
- Opposite angles in a cyclic quadrilateral add up to 180°.
- 圆内接四边形的对角之和为 180°。
- Angles in the same segment are equal.
- 同弧或等弧所对的圆周角相等。
- The radius is perpendicular to the tangent at the point of contact.
- 半径垂直于切点处的切线。
- Tangents from an external point are equal in length.
- 从圆外一点引出的两条切线长度相等。
These circle theorems are often tested in multi-step geometry questions. Always draw the radius to meet the tangent at 90° when it helps.
这些圆的性质经常在多步几何题中出现。当有助于解题时，务必作出半径与切线垂直。
5. Perimeter and Area | 周长与面积
Area is the amount of space inside a 2D shape, while perimeter is the total length around the shape.
面积是二维形状内部所包含的空间大小，周长是绕图形一周的总长度。
| Shape / 图形 | Area formula / 面积公式 |
| Rectangle 长方形 | A = l × w 长 × 宽 |
| Triangle 三角形 | A = ½ × base × height 底 × 高 ÷ 2 |
| Parallelogram 平行四边形 | A = base × height 底 × 高 |
| Trapezium 梯形 | A = ½ × (a + b) × h 上底与下底之和 × 高 ÷ 2 |
| Circle 圆 | A = πr² 圆周率 × 半径² |
The circumference of a circle is C = 2πr or C = πd.
圆的周长公式为 C = 2πr 或 C = πd。
For compound shapes, split them into simple shapes, calculate each area separately, then add or subtract.
对于组合图形，将其拆分为简单图形，分别计算面积，然后相加或相减。
6. Volume and Surface Area | 体积与表面积
Volume measures the space inside a 3D solid. Surface area is the total area of all faces.
体积度量三维立体内部的空间，表面积是所有面的总面积。
Key formulas for prisms and cylinders:
棱柱与圆柱的关键公式：
Volume of a prism = Area of cross-section × Length
棱柱体积 = 横截面积 × 长度
Volume of a cylinder = πr²h
圆柱体积 = πr²h
| Solid / 立体 | Volume / 体积 | Surface area / 表面积 |
| Cuboid 长方体 | l × w × h | 2(lw + wh + hl) |
| Sphere 球体 | ⁴⁄₃ πr³ | 4πr² |
| Cone 圆锥 | ⅓ πr²h | πr² + πrl (l = slant height) |
| Pyramid 棱锥 | ⅓ × base area × height | Sum of all faces 各面面积之和 |
Always check whether the slant height, not the vertical height, is required for the curved surface area of a cone.
计算圆锥侧面积时，要看清题目给出的是母线长（斜高）而不是垂直高。
7. Pythagoras' Theorem | 勾股定理
For a right-angled triangle with hypotenuse c and legs a and b, Pythagoras' theorem states:
对于斜边为 c、两条直角边为 a 和 b 的直角三角形，勾股定理如下：
a² + b² = c²
a² + b² = c²
The hypotenuse is always the side opposite the right angle and is the longest side.
斜边永远是直角所对的边，也是最长边。
Example: Find the hypotenuse when a = 5 and b = 12. c² = 25 + 144 = 169, so c = √169 = 13.
例：已知 a = 5，b = 12，求斜边。c² = 25 + 144 = 169，所以 c = √169 = 13。
Pythagoras' theorem can be used to find a missing side in any right-angled triangle, including those inside 3D solids.
勾股定理可用于求任何直角三角形中的未知边，包括三维立体内部形成的直角三角形。
8. Similarity and Congruence | 相似与全等
Two shapes are congruent if they are exactly the same size and shape. They are similar if they have the same shape but a different size.
两个图形如果大小和形状完全相同，则它们全等。如果形状相同但大小不同，则它们相似。
For similar triangles, corresponding angles are equal and corresponding sides are in the same ratio.
对于相似三角形，对应角相等，对应边成比例。
Scale factor k = (new length) / (original length)
比例系数 k = 新长度 ÷ 原长度
- Length scale factor = k
- 长度比例因子 = k
- Area scale factor = k²
- 面积比例因子 = k²
- Volume scale factor = k³
- 体积比例因子 = k³
Congruence tests for triangles: SSS, SAS, ASA (or AAS) and RHS.
三角形全等的判定方法：边边边（SSS）、边角边（SAS）、角边角（ASA 或 AAS）、直角边斜边（RHS）。
9. Transformations | 几何变换
There are four main transformations on the coordinate plane.
坐标平面中有四种基本变换。
- Translation moves every point by the same vector (x, y).
- 平移：每个点沿同一个向量 (x, y) 移动。
- Reflection flips a shape across a mirror line.
- 反射：图形关于某条对称轴对称翻转。
- Rotation turns a shape about a centre by a given angle.
- 旋转：图形绕某个中心旋转给定的角度。
- Enlargement changes the size using a scale factor and centre of enlargement.
- 放缩/放大：以某个放大中心按比例系数改变图形大小。
When describing a transformation, give the full details: the type, the vector or line/centre/scale factor, and the angle.
描述变换时，必须给出完整信息：变换类型、平移向量或对称轴/旋转中心/放大中心、旋转角度或比例系数。
10. Bearings and Scale Drawings | 方位角与比例作图
A bearing is an angle measured clockwise from north. It is always written with three digits, for example 047°.
方位角是从正北方向顺时针量得的角，通常用三位数表示，例如 047°。
To solve bearing problems, draw a diagram, mark the north line, and use angle facts to find the required angle.
解方位角问题时，先画图，标出正北方向，再利用角度性质求出所需角度。
Scale drawings use a scale such as 1 cm : 2 m. Always measure carefully and convert using the scale.
比例作图使用类似“1 cm : 2 m”的比例尺。测量时务必精确，并按比例尺换算。
The north component is 5 × cos 60° = 2.5 km; the east component is 5 × sin 60° ≈ 4.33 km.
例如：一艘船沿方位角 060° 航行 5 km。正北分量为 5 × cos60° = 2.5 km；正东分量约为 5 × sin60° ≈ 4.33 km。
Remember to present bearings as three-figure angles and always use a protractor in practical questions.
切记方位角要用三位数表示，实际题目中要使用量角器。
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