3D Shapes and Orthographic Projections — 三维图形与正投影视图

Introduction: What Are 3D Shapes? — 简介：什么是三维图形？

In geometry, we distinguish between two-dimensional (2D) shapes and three-dimensional (3D) shapes. While 2D shapes such as squares, rectangles, triangles, and circles exist on a flat plane with only length and width, 3D shapes occupy space and have three dimensions: length, width, and height. In the Cambridge Lower Secondary Mathematics curriculum (Stage 8), students build on their understanding of 2D geometry to explore the properties of common 3D solids, including cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres. This unit lays the foundation for later work on surface area, volume, and spatial reasoning at IGCSE level.

在几何学中，我们区分二维图形和三维图形。二维图形如正方形、长方形、三角形和圆存在于平面上，只有长和宽；而三维图形占据空间，具有三个维度：长、宽和高。在剑桥初中数学课程（Stage 8）中，学生在二维几何理解的基础上，探索常见三维立体的性质，包括正方体、长方体、棱柱、圆柱、棱锥、圆锥和球体。本单元为后续 IGCSE 阶段的表面积、体积和空间推理学习奠定基础。

Key Properties of 3D Shapes: Faces, Edges, and Vertices — 三维图形的关键性质：面、棱和顶点

Every three-dimensional solid can be described using three key structural components. A face is a flat surface that forms part of the boundary of the solid. An edge is the line segment where two faces meet. A vertex (plural: vertices) is a corner point where three or more edges intersect. Understanding these components allows us to systematically analyse any 3D shape and calculate its geometric properties. For a cuboid, for instance, the faces are rectangular surfaces, the edges are the line segments joining the vertices, and the vertices are the eight corner points of the shape.

每个三维立体都可以用三个关键结构部件来描述。面是构成立体边界一部分的平坦表面。棱是两个面相交的线段。顶点是三条或更多棱相交的角点。理解这些组成部分使我们能够系统地分析任何三维图形并计算其几何性质。例如，对于长方体，面是矩形表面，棱是连接顶点的线段，顶点是图形的八个角点。

Euler’s Formula: A Fundamental Relationship — 欧拉公式：一个基本关系

One of the most elegant discoveries in geometry is Euler’s Formula, named after the Swiss mathematician Leonhard Euler. For any convex polyhedron (a 3D solid whose faces are all flat polygons and whose surface curves inward nowhere), the number of faces (F), vertices (V), and edges (E) are related by a simple equation: F + V – E = 2. This means that if you add the number of faces and vertices and subtract the number of edges, the result is always 2, regardless of the shape’s size or complexity. This relationship holds true for all convex polyhedra, from a simple tetrahedron with just four triangular faces to a complex icosahedron with twenty triangular faces.

几何学中最优雅的发现之一是欧拉公式，以瑞士数学家莱昂哈德·欧拉命名。对于任何凸多面体（所有面都是平坦多边形的三维立体，且表面没有向内弯曲），面数 (F)、顶点数 (V) 和棱数 (E) 由一个简单方程关联：F + V – E = 2。这意味着无论图形的大小或复杂程度如何，如果加面数和顶点数再减去棱数，结果总是 2。这个关系对所有凸多面体都成立，从只有四个三角形面的简单四面体到有二十个三角形面的复杂二十面体。

Applying Euler’s Formula: Worked Examples — 欧拉公式的应用：例题解析

Let us verify Euler’s Formula with some common 3D shapes. Consider a cube. A cube has 6 faces (all squares), 8 vertices (the corners), and 12 edges (the line segments connecting the vertices). Substituting into the formula: 6 + 8 – 12 = 2. The formula holds. Now consider a square-based pyramid. This shape has 5 faces (4 triangular faces and 1 square base), 5 vertices (4 at the base corners and 1 at the apex), and 8 edges (4 around the base and 4 sloping up to the apex). Checking: 5 + 5 – 8 = 2. Once again, Euler’s Formula is confirmed. Students should practise verifying this formula for various polyhedra, including prisms with different polygonal bases.

让我们用一些常见的三维图形来验证欧拉公式。考虑一个正方体。正方体有 6 个面（全部是正方形）、8 个顶点（角）和 12 条棱（连接顶点的线段）。代入公式：6 + 8 – 12 = 2。公式成立。现在考虑一个四棱锥。这个图形有 5 个面（4 个三角形面和 1 个正方形底面）、5 个顶点（底面 4 个角点加锥顶 1 个）和 8 条棱（底面 4 条加上通向锥顶的 4 条）。验证：5 + 5 – 8 = 2。欧拉公式再次被确认。学生应该练习为各种多面体验证这个公式，包括具有不同多边形底面的棱柱。

The Cuboid: A Detailed Case Study — 长方体：详细案例分析

The cuboid (or rectangular prism) is one of the most commonly encountered 3D shapes in everyday life. Think of a shoebox, a brick, or a textbook. The cuboid has exactly six rectangular faces arranged in three pairs of opposite faces. Opposite faces are congruent, meaning they have exactly the same dimensions. A cuboid has twelve edges, which can be grouped into three sets of four parallel edges of equal length. These correspond to the three dimensions: the length, the width, and the height. The cuboid also has eight vertices, each formed by the intersection of three edges that are mutually perpendicular. When all six faces are squares and all edges are equal in length, the cuboid becomes a special case known as a cube.

长方体是日常生活中最常见的三维图形之一。想想鞋盒、砖块或课本。长方体恰好有六个矩形面，排列成三对相对的面。相对的面是全等的，意味着它们具有完全相同的尺寸。长方体有十二条棱，可以分成三组每组四条等长且平行的棱。它们对应三个维度：长、宽和高。长方体还有八个顶点，每个顶点由三条相互垂直的棱相交形成。当所有六个面都是正方形且所有棱长度相等时，长方体就变成了一个特殊情况 – 正方体。

Orthographic Projections: Seeing 3D in 2D — 正投影：在二维中看三维

One of the most important practical skills in geometry is the ability to represent a three-dimensional object accurately on a two-dimensional surface. This is achieved through orthographic projection, a technique widely used in engineering, architecture, and design. An orthographic projection shows what an object looks like when viewed from a specific direction, typically the top, the front, and the side. Each of these views is a 2D representation that preserves the true shape and dimensions of the faces visible from that direction. Unlike perspective drawings that show depth through converging lines, orthographic projections use parallel projection lines, ensuring that parallel edges in the object remain parallel in the drawing.

几何学中最重要的实用技能之一是能够在二维表面上准确表示三维物体。这通过正投影来实现，这是一种在工程、建筑和设计中广泛使用的技术。正投影显示从特定方向（通常是顶部、正面和侧面）观察物体时所看到的内容。每个视图都是一个二维表示，保留了从该方向可见面的真实形状和尺寸。与通过汇聚线显示深度的透视图不同，正投影使用平行投影线，确保物体中的平行边在绘图时保持平行。

Drawing Top, Front, and Side Views — 绘制俯视图、正视图和侧视图

To draw the orthographic views of a 3D solid, follow these systematic steps. First, the top view (also called the plan view) shows what you would see if you looked directly down from above. For a cuboid, the top view is a rectangle with dimensions equal to the length and width of the solid. Second, the front view (also called the front elevation) shows what you see when looking at the solid from directly in front. This is a rectangle with dimensions equal to the length and height. Third, the side view (also called the side elevation) shows what you see from the side. For a cuboid, this is a rectangle with dimensions equal to the width and height. When drawing the three views, it is standard practice to align them: the top view is placed directly above the front view, and the side view is placed to the right of the front view.

要绘制三维立体的正投影视图，请遵循以下系统步骤。首先，俯视图显示从正上方垂直向下看所看到的内容。对于长方体，俯视图是一个矩形，其尺寸等于立体的长和宽。其次，正视图显示从正前方看立体时所看到的内容。这是一个矩形，尺寸等于长和高。第三，侧视图显示从侧面看时所看到的内容。对于长方体，这是一个矩形，尺寸等于宽和高。绘制三个视图时，标准做法是将它们对齐：俯视图放在正视图的正上方，侧视图放在正视图的右侧。

Scale Drawings in Orthographic Projections — 正投影中的比例尺绘图

When objects are too large to draw at their actual size, or too small to show detail clearly, we use a scale. A scale is a ratio that relates the dimensions in the drawing to the actual dimensions of the object. For example, a scale of 1:5 means that every 1 cm on the drawing represents 5 cm on the actual object. To convert actual dimensions to drawing dimensions, divide the real measurement by the scale factor. If a cuboid measures 35 cm in length, 15 cm in width, and 10 cm in height, using a scale of 1:5 gives drawing dimensions of 7 cm by 3 cm by 2 cm respectively. Always label the scale clearly on your drawing. Common scales include 1:2, 1:5, 1:10, and 1:100, the last being standard for architectural floor plans.

当物体太大无法按实际尺寸绘制，或太小难以显示细节时，我们使用比例尺。比例尺是一个比率，将绘图中的尺寸与物体的实际尺寸相关联。例如，比例尺 1:5 意味着图纸上的每 1 厘米代表实际物体上的 5 厘米。要将实际尺寸转换为绘图尺寸，将实际测量值除以比例因子。如果一个长方体长 35 厘米，宽 15 厘米，高 10 厘米，使用比例尺 1:5 得到的绘图尺寸分别为 7 厘米、3 厘米和 2 厘米。始终在图纸上清楚地标注比例尺。常见比例尺包括 1:2、1:5、1:10 和 1:100，其中 1:100 是建筑平面图的标准比例。

Worked Example: Cuboid with Scale 1:5 — 例题：长方体比例尺 1:5 绘图

Let us work through a complete example similar to the one found in Cambridge Lower Secondary Mathematics Stage 8. A cuboid has dimensions: length = 35 cm, width = 15 cm, height = 10 cm. Using a scale of 1:5, calculate the drawing dimensions and sketch the three orthographic views. The scale means 1 cm on paper represents 5 cm in reality. Drawing length: 35 divided by 5 = 7 cm. Drawing width: 15 divided by 5 = 3 cm. Drawing height: 10 divided by 5 = 2 cm. The top view is a rectangle measuring 7 cm by 3 cm. The front view is a rectangle measuring 7 cm by 2 cm. The side view is a rectangle measuring 3 cm by 2 cm. Always use a ruler, sharp pencil, and set square for accurate construction. Dimensions should be placed outside the views with extension lines.

让我们完整地做一个例题，类似于剑桥初中数学 Stage 8 教材中的例子。一个长方体尺寸为：长 = 35 厘米，宽 = 15 厘米，高 = 10 厘米。使用比例尺 1:5，计算绘图尺寸并绘制三个正投影视图。比例尺意味着纸上 1 厘米代表实际 5 厘米。绘图长度：35 除以 5 = 7 厘米。绘图宽度：15 除以 5 = 3 厘米。绘图高度：10 除以 5 = 2 厘米。俯视图是一个 7 厘米 × 3 厘米的矩形。正视图是一个 7 厘米 × 2 厘米的矩形。侧视图是一个 3 厘米 × 2 厘米的矩形。始终使用直尺、削尖的铅笔和三角板来精确作图。尺寸应标注在视图之外并添加尺寸延伸线。

Beyond Cuboids: Orthographic Projections of Other 3D Shapes — 超越长方体：其他三维图形的正投影

While cuboids produce simple rectangular projections, other 3D shapes yield more interesting views. Consider a cylinder. The top view of a cylinder is a circle, equal in diameter to the cylinder’s circular base. The front view is a rectangle whose width equals the diameter of the base and whose height equals the cylinder’s height. The side view is identical to the front view. Now consider a square-based pyramid. The top view shows a square (the base) with the apex appearing as a point at the centre. The front view is an isosceles triangle with its base equal to the side length of the square base and its height equal to the pyramid’s height. The side view is also an isosceles triangle, but its base equals the width of the square base. For more complex shapes, hidden edges are shown as dashed lines in orthographic projections.

虽然长方体产生简单的矩形投影，但其他三维图形会产生更有趣的视图。考虑一个圆柱体。圆柱体的俯视图是一个圆，直径等于圆柱体的圆形底面直径。正视图是一个矩形，宽度等于底面直径，高度等于圆柱体的高度。侧视图与正视图相同。现在考虑一个四棱锥。俯视图显示一个正方形（底面），锥顶在中心显示为一个点。正视图是一个等腰三角形，底边等于正方形底面的边长，高度等于棱锥的高度。侧视图也是一个等腰三角形，但底边等于正方形底面的宽。对于更复杂的形状，隐藏的边在正投影中用虚线表示。

Nets of 3D Shapes: Unfolding Solids — 三维图形的展开图：展开立体

A net of a 3D shape is a two-dimensional pattern that can be folded to create the solid. Understanding nets helps students visualise the relationship between 2D and 3D shapes. The net of a cube consists of six squares arranged in a cross-shaped pattern. A cuboid’s net contains six rectangles arranged so that each rectangle touches its neighbours along edges that will become the edges of the solid. When drawing nets, it is essential to include tabs for gluing and to ensure that the correct faces are adjacent. Not all arrangements of the faces produce valid nets. For a cube, there are exactly eleven distinct nets, known as the hexominoes that can fold into a cube. Exploring nets reinforces understanding of faces, edges, and how they connect in three-dimensional space.

三维图形的展开图是一个可以折叠成立体的二维图形模板。理解展开图有助于学生可视化二维和三维图形之间的关系。正方体的展开图由排列成十字形的六个正方形组成。长方体的展开图包含六个矩形，排列方式使得每个矩形沿将成为立体棱的边与相邻矩形接触。绘制展开图时，必须包含粘贴边，并确保正确的面相邻。并非所有的面排列方式都能产生有效的展开图。对于正方体，恰好有十一种不同的展开图，称为可以折叠成正方体的六格骨牌。探索展开图可以加深对面、棱以及它们在三维空间中如何连接的理解。

Common Mistakes and How to Avoid Them — 常见错误及如何避免

Students frequently encounter several pitfalls when working with 3D shapes and orthographic projections. One common mistake is confusing the number of edges with the number of vertices, or miscounting faces by forgetting the base or the top of the solid. Always count systematically: first all visible faces, then check for hidden ones. Another frequent error is using the wrong dimensions for the side view; remember that the side view width is the solid’s width, not its length. When using a scale, students often forget to divide all three dimensions by the scale factor. A third mistake involves alignment: the three orthographic views must be correctly aligned, with the top view directly above the front view and the side view horizontally level with the front view. Misaligned views make it impossible to read the drawing correctly. Lastly, always check that your answer satisfies Euler’s Formula as a verification step.

学生在处理三维图形和正投影时经常遇到几个陷阱。一个常见错误是混淆棱数和顶点数，或忘记计算底面或顶面而导致面数计数错误。始终系统地计数：先数所有可见的面，再检查隐藏的面。另一个常见错误是侧视图使用了错误的尺寸；记住侧视图的宽度是立体的宽，而不是长。使用比例尺时，学生经常忘记将所有三个维度都除以比例因子。第三个错误涉及对齐：三个正投影视图必须正确对齐，俯视图在正视图的正上方，侧视图与正视图水平对齐。未对齐的视图使得无法正确解读图纸。最后，始终检查你的答案是否满足欧拉公式作为验证步骤。

Practice Questions: Test Your Understanding — 练习题目：测试你的理解

Question 1: A triangular prism has 5 faces, 6 vertices, and 9 edges. Verify Euler’s Formula for this solid. Question 2: A cuboid measures 40 cm in length, 20 cm in width, and 15 cm in height. Using a scale of 1:4, determine the dimensions of the top view, front view, and side view in centimetres. Question 3: A cylinder has a radius of 3 cm and a height of 8 cm. Describe what the top view and the front view look like, and give their dimensions. Question 4: How many distinct nets does a cube have? Can you draw at least three different ones? Question 5: A solid has 7 faces and 10 vertices. Using Euler’s Formula, calculate how many edges it has. What might this solid be?

题目 1：一个三棱柱有 5 个面、6 个顶点和 9 条棱。验证欧拉公式对此立体是否成立。题目 2：一个长方体长 40 厘米、宽 20 厘米、高 15 厘米。使用比例尺 1:4，确定俯视图、正视图和侧视图的尺寸（单位：厘米）。题目 3：一个圆柱体半径 3 厘米、高 8 厘米。描述俯视图和正视图是什么样的，并给出它们的尺寸。题目 4：正方体有多少种不同的展开图？你能画出至少三种吗？题目 5：一个立体有 7 个面和 10 个顶点。使用欧拉公式计算它有多少条棱。这个立体可能是什么？

Systematic Methods for Counting Faces, Vertices, and Edges — 系统计数面、顶点和棱的方法

When a 3D shape is complex, simply looking at a diagram and trying to count can lead to mistakes. Cambridge Lower Secondary Mathematics teaches a systematic approach. For counting faces, imagine holding the solid in your hand and rotating it. Start with the base (bottom face), then count the sides going around, then finish with the top face. For a hexagonal prism, this gives: 1 (bottom) + 6 (rectangular sides) + 1 (top) = 8 faces. For counting vertices, start at the bottom layer and work around it, then move to the top layer. For the same hexagonal prism: 6 vertices on the bottom hexagon plus 6 on the top hexagon = 12 vertices. For counting edges, count the edges on the top face, the edges on the bottom face, and then the vertical edges connecting them. For the hexagonal prism: 6 (top) + 6 (bottom) + 6 (vertical) = 18 edges. Verify with Euler’s Formula: 8 + 12 – 18 = 2. This layered counting method works for all prisms and many other polyhedra.

当三维图形复杂时，仅仅看图表并尝试计数可能会导致错误。剑桥初中数学教授一种系统的方法。计数面时，想象你手持这个立体并旋转它。从底面（底部）开始，然后绕一圈数侧面，最后数顶面。对于六棱柱：1（底）+ 6（矩形侧面）+ 1（顶）= 8 个面。计数顶点时，从底层开始绕一圈，然后移到顶层。对于同一个六棱柱：底层六边形上有 6 个顶点，加上顶层六边形上的 6 个顶点 = 12 个顶点。计数棱时，数顶面的棱、底面的棱，然后数连接它们的竖直棱。对于六棱柱：6（顶）+ 6（底）+ 6（竖直）= 18 条棱。用欧拉公式验证：8 + 12 – 18 = 2。这种分层计数方法适用于所有棱柱和许多其他多面体。

Prisms: A Special Class of Three-Dimensional Shapes — 棱柱：一类特殊的三维图形

A prism is a three-dimensional solid with two identical, parallel polygonal bases connected by rectangular lateral faces. The shape of the base gives the prism its name: a triangular prism has triangular bases, a pentagonal prism has pentagonal bases, and a hexagonal prism has hexagonal bases. Prisms follow a consistent pattern. If the base has n sides, the prism has n + 2 faces (n lateral rectangles plus 2 bases), 2n vertices (n on each base), and 3n edges (n on each base plus n lateral edges). The cross-section of a prism, taken parallel to the bases, is always the same shape and size. This property is what makes a prism a prism, distinguishing it from pyramids (which taper to a point) and other solids. In the Cambridge Lower Secondary Mathematics curriculum, students are expected to recognise and describe prisms by their cross-sectional shape.

棱柱是一个具有两个相同、平行的多边形底面，并由矩形侧面连接的三维立体。底面的形状决定了棱柱的名称：三棱柱有三角形底面，五棱柱有五边形底面，六棱柱有六边形底面。棱柱遵循一个一致的模式。如果底面有 n 条边，则棱柱有 n + 2 个面（n 个矩形侧面加上 2 个底面）、2n 个顶点（每个底面 n 个）和 3n 条棱（每个底面 n 条加上 n 条侧棱）。平行于底面的棱柱横截面始终具有相同的形状和大小。这一性质使棱柱区别于棱锥（向一点收缩）和其他立体。在剑桥初中数学课程中，学生需要根据棱柱的横截面形状来识别和描述棱柱。

Isometric Drawing: An Alternative to Orthographic Projection — 等距绘图：正投影的替代方案

While orthographic projections show separate 2D views, isometric drawing offers a single 3D-like representation on a 2D surface. In an isometric drawing, the three axes are equally inclined at 120 degrees to each other, and distances along these axes are drawn to the same scale (hence “iso” meaning equal and “metric” meaning measure). Unlike true perspective drawings where objects appear smaller with distance, isometric drawings preserve relative proportions. For Cambridge Lower Secondary Mathematics Stage 8, students learn to sketch simple solids on isometric dot paper. To draw a cuboid isometrically: draw the front vertical edge, then draw the bottom edges at 30 degrees to the horizontal, then draw the top edges parallel to the corresponding bottom edges, and finally complete the vertical edges. The advantage of isometric drawing is that it shows three faces at once in a single view, making it easier to visualise the overall shape. The disadvantage is that angles and circles are distorted, which is why orthographic projection remains essential for precise technical drawings.

虽然正投影显示分离的二维视图，但等距绘图在二维表面上提供了单一的三维立体表示。在等距绘图中，三条轴彼此等倾 120 度，沿这些轴的距离以相同的比例绘制（因此”iso”表示相等，”metric”表示度量）。与物体随距离变小的真实透视图不同，等距绘图保持相对比例。在剑桥初中数学 Stage 8 中，学生学会在等距点阵纸上绘制简单的立体图形。要等距绘制一个长方体：先画前竖直边，然后以水平方向 30 度角画底边，再画出与对应底边平行的顶边，最后完成竖直边。等距绘图的优点是它在单个视图中同时显示三个面，更容易可视化整体形状。缺点是角度和圆会被扭曲，这就是为什么正投影对于精确的技术绘图仍然不可或缺。

Real-World Applications of 3D Geometry and Orthographic Projection — 立体几何与正投影的实际应用

The skills developed in this topic extend far beyond the classroom. Architects use orthographic projections to produce floor plans (top view), elevations (front and side views), and sections for every building they design. Engineers rely on these drawings to communicate precise specifications for manufacturing components, from smartphone cases to aircraft parts. Product designers sketch orthographic views to specify the exact dimensions of consumer goods before creating prototypes. In the medical field, CT scans and MRI images are essentially digital orthographic slices through the human body. Even in video game design and computer animation, understanding how 3D objects project onto a 2D screen is fundamental. The ability to read and interpret orthographic drawings is a valued skill in many technical careers, and the Cambridge Lower Secondary Mathematics programme introduces these concepts at an accessible level to build students’ spatial reasoning abilities from an early age.

本主题培养的技能远远超出课堂。建筑师使用正投影来制作他们设计的每栋建筑的平面图（俯视图）、立面图（正视图和侧视图）以及剖面图。工程师依赖这些图纸来传达制造部件（从手机壳到飞机零件）的精确规格。产品设计师绘制正投影视图以指定消费品的确切尺寸，然后再制作原型。在医学领域，CT 扫描和 MRI 图像本质上是人体内部的数字正投影切片。即使在电子游戏设计和计算机动画中，理解三维物体如何投影到二维屏幕上也是基础性的。阅读和解读正投影图纸的能力是许多技术职业中受重视的技能，剑桥初中数学课程在合适的水平介绍这些概念，以从早期开始培养学生的空间推理能力。

Summary: 3D Shapes and Orthographic Projections — 总结：三维图形与正投影

This topic introduces the fundamental concepts of three-dimensional geometry required in the Cambridge Lower Secondary Mathematics Stage 8 curriculum. Students learn to identify and count the faces, vertices, and edges of common 3D solids, apply Euler’s Formula (F + V – E = 2) as a powerful checking tool, and draw accurate orthographic projections from the top, front, and side views. The cuboid serves as the primary example throughout, with extensions to prisms, pyramids, cylinders, and other polyhedra. Scale drawings tie these geometric skills to real-world applications in engineering, architecture, and technical design. Mastery of these concepts provides a solid foundation for the more advanced topics of surface area, volume, and trigonometry that students will encounter at IGCSE level and beyond.

本主题介绍了剑桥初中数学 Stage 8 课程所需的立体几何基本概念。学生学习识别和计数常见三维立体的面、顶点和棱，运用欧拉公式 (F + V – E = 2) 作为强大的验证工具，并从俯视、正视和侧视三个方向绘制准确的正投影视图。长方体贯穿始终作为主要例子，并扩展到棱柱、棱锥、圆柱和其他多面体。比例尺绘图将这些几何技能与工程、建筑和技术设计中的实际应用联系起来。掌握这些概念为学生今后在 IGCSE 及更高阶段遇到表面积、体积和三角学等更高级主题奠定坚实基础。

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