📚 Properties of 3D Shapes | 三维图形的性质
Understanding the properties of three-dimensional shapes is a cornerstone of geometry in the IB Mathematics programme. From identifying faces, edges and vertices to calculating surface areas and volumes, these spatial concepts help students develop logical reasoning and visualisation skills that are essential for both pure and applied mathematics.
理解三维图形的性质是IB数学课程中几何学的基石。从识别面、棱和顶点到计算表面积和体积，这些空间概念有助于学生培养逻辑推理和可视化能力，这对于纯数学和应用数学都至关重要。
1. Introduction to 3D Shapes | 三维图形导论
Three-dimensional shapes, often called solids, occupy space and have three measurable dimensions: length, width and height. Unlike two-dimensional figures, they cannot be fully represented on a flat surface without some distortion and they possess volume – the amount of space they enclose.
三维图形通常称为立体，占据空间并具有三个可测量的维度：长度、宽度和高度。与二维图形不同，它们无法在没有变形的情况下完全在平面上表示出来，并且它们拥有体积——即所包含的空间量。
Every solid can be classified as either a polyhedron (bounded entirely by flat polygonal faces) or a non-polyhedron (involving curved surfaces). Cubes, pyramids and prisms are polyhedra, while cylinders, cones and spheres are examples of curved solids.
每个立体都可以归类为多面体（完全由平坦的多边形面构成）或非多面体（包含曲面）。立方体、棱锥和棱柱是多面体，而圆柱、圆锥和球体是曲面立体的例子。
2. Faces, Edges and Vertices | 面、棱和顶点
A face is a flat or curved surface that forms part of the boundary of a solid. In a polyhedron, every face is a polygon. For instance, a rectangular prism has six rectangular faces.
面是构成立体边界的平坦或曲面表面。在多面体中，每个面都是多边形。例如，一个长方体有六个矩形面。
An edge is the line segment formed where two faces meet. A cube has 12 edges, each created by the intersection of two square faces.
棱是两个面相交形成的线段。立方体有12条棱，每条棱由两个正方形面相交形成。
A vertex (plural vertices) is a corner point where three or more edges join. A triangular pyramid has 4 vertices – the three base corners and the apex.
顶点是三条或更多条棱汇合的角点。一个三棱锥有4个顶点——三个底角和一个顶点。
Counting faces, edges and vertices accurately is the first step in analysing any polyhedron and is directly linked to one of the most elegant formulas in geometry.
准确计数面、棱和顶点是分析任何多面体的第一步，并与几何学中最优雅的公式之一直接相关。
3. Euler’s Formula for Polyhedra | 多面体的欧拉公式
For any convex polyhedron, the numbers of faces (F), vertices (V) and edges (E) are related by Euler’s formula, a cornerstone of topological geometry.
对于任何凸多面体，面数（F）、顶点数（V）和棱数（E）由欧拉公式关联，这是拓扑几何学的基石。
F + V − E = 2
ces and 6 edges, giving 4+4−6=2. The formula even helps detect errors when sketching nets.
这一简单关系对所有凸多面体都成立。用立方体验证：F=6，V=8，E=12，所以6+8−12=2。四面体有4个面、4个顶点和6条棱，得出4+4−6=2。该公式甚至有助于在画展开图时发现错误。
| Polyhedron | F | V | E | F+V−E |
|---|---|---|---|---|
| Tetrahedron | 4 | 4 | 6 | 2 |
| Cube | 6 | 8 | 12 | 2 |
| Octahedron | 8 | 6 | 12 | 2 |
| Square pyramid | 5 | 5 | 8 | 2 |
| Pentagonal prism | 7 | 10 | 15 | 2 |
The table confirms that F+V−E is consistently 2, reinforcing the robustness of Euler’s formula across various polyhedral shapes.
该表证实了F+V−E始终为2，强化了欧拉公式在不同多面体形状中的稳健性。
4. Types of Polyhedra: Prisms | 多面体类型：棱柱
A prism is a polyhedron with two parallel, congruent polygonal bases connected by rectangular lateral faces. The shape of the base gives the prism its name – a triangular prism has triangular bases, and a hexagonal prism has hexagonal bases.
棱柱是一种多面体，具有两个平行且全等的多边形底面，并由矩形侧面连接。底面的形状决定了棱柱的名称——三棱柱有三角形底面，六棱柱有六边形底面。
oss-sectional area, a property that simplifies volume calculation: V = base area × height.
与棱柱底面平行的所有截面都与底面相同。这种一致性意味着棱柱的截面积恒定，这一性质简化了体积计算：V = 底面积 × 高。
Right prisms have lateral edges perpendicular to the base; oblique prisms lean over. In IB examinations, attention is usually focused on right prisms, where lateral faces are rectangles.
直棱柱的侧棱垂直于底面；斜棱柱是倾斜的。在IB考试中，通常关注直棱柱，其侧面为矩形。
5. Types of Polyhedra: Pyramids | 多面体类型：棱锥
A pyramid consists of a polygonal base and triangular lateral faces that meet at a common vertex called the apex. Like prisms, pyramids are named after their base polygon – a square pyramid has a square base, a pentagonal pyramid has a pentagonal base.
棱锥由一个多边形底面和汇聚于一个共同顶点（称为锥顶）的三角形侧面组成。与棱柱类似，棱锥也根据底面多边形命名——四棱锥有一个正方形底面，五棱锥有一个五边形底面。
Pyramids do not have constant cross-sectional area; the area decreases linearly as you move from the base to the apex. As a result, the volume of any pyramid is exactly one-third the product of its base area and vertical height.
棱锥的截面积不是恒定的；从底面到锥顶，面积线性减小。因此，任何棱锥的体积恰好是其底面积与垂直高度乘积的三分之一。
V = 1/3 × base area × height
A regular pyramid has a regular polygon base and lateral faces that are congruent isosceles triangles, making symmetry properties easier to analyse.
正棱锥的底面是一个正多边形，侧面是全等的等腰三角形，这使得对称性更容易分析。
6. Platonic Solids | 柏拉图立体
Platonic solids are regular convex polyhedra in which every face is the same regular polygon and the same number of faces meet at each vertex. There are exactly five such solids: tetrahedron, cube, octahedron, dodecahedron and icosahedron.
柏拉图立体是正凸多面体，每个面都是相同的正多边形，且每个顶点连接的面数相同。这样的立体恰好有五种：正四面体、立方体、正八面体、正十二面体和正二十面体。
0 equilateral triangles) are more complex but satisfy the same Euler relationship F+V−E=2.
正四面体（4个等边三角形）、立方体（6个正方形）和正八面体（8个等边三角形）常在IB几何课题中探讨。正十二面体（12个正五边形）和正二十面体（20个等边三角形）更复杂，但同样满足欧拉关系F+V−E=2。
Platonic solids are symmetrical in many ways: they have multiple axes of rotational symmetry and reflection planes, making them fundamental in the study of spatial symmetry.
柏拉图立体在多个方面具有对称性：它们有多条旋转对称轴和反射平面，这使它们成为空间对称研究的基础。
7. Curved Surface Solids: Cylinders, Cones and Spheres | 曲面立体：圆柱、圆锥和球
A cylinder has two parallel circular bases and a curved lateral surface. It can be thought of as a prism with a circular base, so its volume is πr²h and its surface area includes the two base circles plus the curved rectangle that wraps around.
圆柱体有两个平行的圆形底面和一个弯曲的侧面。它可以看作是一个底面为圆形的“棱柱”，因此其体积为πr²h，表面积包括两个底面圆加上环绕的弯曲矩形。
e area involves a sloping height calculation: SA = πr² + πrl, where l is the slant height.
圆锥有一个圆形底面和一个逐渐收缩到一个顶点的曲面。体积是1/3 × πr²h，类似棱锥，但其表面积涉及斜高计算：SA = πr² + πrl，其中l是斜高。
A sphere is the set of all points at a fixed distance from a centre. It has no faces, edges or vertices. The surface area is 4πr² and volume is (4/3)πr³, both of which are key formulas in IB Mathematics.
球体是到中心距离固定的所有点的集合。它没有面、棱或顶点。表面积为4πr²，体积为(4/3)πr³，这两个都是IB数学中的关键公式。
8. Nets of 3D Shapes | 三维图形的展开图
A net is a two-dimensional representation that can be folded to form a three-dimensional shape. Students often use nets to investigate surface area, visualise the arrangement of faces and check whether a given construction is possible.
展开图是一种可以折叠形成三维图形的二维表示。学生常用展开图来研究表面积、可视化面的排列，并检验某个构造是否可能。
For a cube, there are 11 distinct nets, all consisting of six squares arranged edge-to-edge. Not every arrangement of six squares forms a cube net – some will overlap or leave gaps when folded.
立方体有11种不同的展开图，全部由六个以边相连的正方形组成。并非每种六个正方形的排列都能形成一个立方体展开图——有些在折叠时会重叠或留下空隙。
Nets of pyramids and prisms can be constructed by drawing the base and attaching the lateral faces around its perimeter, paying attention to edge lengths that must match.
棱锥和棱柱的展开图可以通过画出底面并在其周边附上侧面来构建，并注意棱长必须匹配。
9. Surface Area Properties | 表面积性质
The surface area of a solid is the total area of all its faces or curved surfaces. For polyhedra, it is simply the sum of the areas of individual polygons; for curved solids, it combines circular areas and generated surfaces from revolving a line.
立体的表面积是其所有面或曲面的总面积。对于多面体，它就是各个多边形面积的总和；对于曲面立体，则结合了圆面积和由线段旋转生成的曲面。
rectangular prism with length l, width w and height h, this becomes SA = 2(lw + lh + wh).
棱柱的表面积可通过SA = 2B + Ph计算，其中B为底面积，P为底面周长，h为高。对于长为l、宽为w、高为h的长方体，这变为SA = 2(lw + lh + wh)。
Understanding how surface area formulas are derived helps IB learners extend them to composite solids and solve optimisation problems, such as minimising material for a fixed volume.
理解表面积公式的推导方式有助于IB学习者将其扩展到组合立体，并解决优化问题，例如在固定体积下使材料最少。
10
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图