Shape and Space 7 | 形状与空间 7

📚 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）

Example: 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

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。

Example: A ship sails 5 km on a bearing of 060°. 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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Shape and Space 7 | 形状与空间 7