📚 Properties of 3D Shapes | 三维图形的性质
In the IB Mathematics curriculum, understanding the properties of three-dimensional (3D) shapes is essential for solving problems involving space, measurement and geometry. This article reviews the key concepts, formulas and strategies you need to master 3D geometry.
在IB数学课程中，理解三维图形的性质对于解决涉及空间、测量和几何的问题至关重要。本文回顾了掌握三维几何所需的关键概念、公式和策略。
1. Vertices, Edges and Faces | 顶点、棱与面
Every 3D solid is composed of three fundamental elements: vertices, edges and faces. A vertex is a point where two or more edges meet; an edge is a line segment where two faces meet; and a face is a flat polygonal surface. For instance, a cube has 8 vertices, 12 edges and 6 square faces.
每个三维立体都由三个基本元素组成：顶点、棱和面。顶点是两条或更多棱相交的点；棱是两个面相交的线段；面是平坦的多边形表面。例如，立方体有8个顶点、12条棱和6个正方形面。
In IB problems, you may be asked to count these elements directly or to use them in Euler’s formula. Be careful with curved surfaces: a cylinder has two circular edges but is not classified as a polyhedron, so the usual edge-count rules for polyhedra may not apply.
在IB题目中，你可能需要直接数出这些元素，或将它们用于欧拉公式。注意曲面：圆柱有两条圆形边，但它不是多面体，因此通常用于多面体的棱数规则可能不适用。
2. Euler’s Formula | 欧拉公式
For any convex polyhedron, Euler’s formula relates the numbers of vertices (V), edges (E) and faces (F):
对于任何凸多面体，欧拉公式将顶点数（V）、棱数（E）和面数（F）联系起来：
V – E + F = 2
A cube has V=8, E=12, F=6, so 8 – 12 + 6 = 2. A triangular pyramid has V=4, E=6, F=4, so 4 – 6 + 4 = 2. This formula provides a powerful check when counting the parts of a polyhedron.
立方体有V=8，E=12，F=6，因此8 – 12 + 6 = 2。三棱锥有V=4，E=6，F=4，因此4 – 6 + 4 = 2。这个公式为检查多面体各部分的计数提供了有力的手段。
3. Prisms and Pyramids | 棱柱与棱锥
A prism has two parallel, congruent bases connected by parallelogram faces. A pyramid has one base and triangular faces that meet at a single apex. Both are named after the shape of their base, for example rectangular prism, hexagonal pyramid.
棱柱有两个平行且全等的底面，由平行四边形侧面连接；棱锥有一个底面，三角形侧面汇于一个顶点。二者均以底面形状命名，如长方体、六棱锥。
For any prism, volume is given by:
对于任意棱柱，体积公式为：
V = A_base × h
where A_base is the base area and h is the perpendicular height. For a pyramid:
其中A_base为底面积，h为垂直高。对于棱锥：
V = ⅓ A_base × h
The surface area is the sum of the areas of all faces: prism area = 2A_base + A_lateral; pyramid area = A_base + A_lateral.
表面积是所有面的面积之和：棱柱表面积 = 2A_base + A_lateral；棱锥表面积 = A_base + A_lateral。
4. Cylinders and Cones | 圆柱与圆锥
A cylinder is a prism with circular bases, and a cone is a pyramid with a circular base. Their volumes and surface areas appear frequently in IB papers.
圆柱是圆形底面的棱柱，圆锥是圆形底面的棱锥。它们的体积和表面积在IB试卷中经常出现。
For a cylinder of radius r and height h:
对于半径为r、高为h的圆柱：
V = πr²h, A_curved = 2πrh, A_total = 2πr² + 2πrh
For a cone of radius r, height h and slant height l:
对于半径为r、高为h、斜高为l的圆锥：
V = ⅓πr²h, A_curved = πrl, A_total = πr² + πrl
The slant height satisfies l² = r² + h² by Pythagoras’ theorem.
斜高满足 l² = r² + h²（勾股定理）。
5. Spheres | 球体
A sphere is the set of all points at a fixed distance r from a centre. Its volume and surface area are essential formulas.
球体是到中心距离恒为r的所有点的集合。它的体积和表面积是必备公式。
V = (4/3)πr³
A = 4πr²
A common exam variant is a hemisphere. The curved surface area is 2πr², and the total surface area including the circular base is 3πr².
一个常见的考点是半球。其曲面积为2πr²，包含圆底面的总表面积为3πr²。
6. Summary Table of Formulas | 公式汇总表
The following table summarises the key volume and surface area formulas for common 3D shapes.
下表总结了常见三维图形的关键体积和表面积公式。
| Shape 图形 | Volume 体积 | Surface Area 表面积 |
|---|---|---|
| Cube 正方体 | a³ | 6a² |
| Rectangular prism 长方体 | abc | 2(ab + bc + ca) |
| Cylinder 圆柱 | πr²h | 2πr² + 2πrh |
| Cone 圆锥 | ⅓πr²h | πr² + πrl |
| Sphere 球体 | (4/3)πr³ | 4πr² |
7. Composite Solids | 复合图形
Composite solids are made by combining or subtracting basic 3D shapes. To find the total volume, add or subtract the constituent volumes. For surface area, consider only the external faces and do not count shared internal surfaces twice.
复合图形由基本三维图形组合或挖去而成。求总体积时，将各部分体积相加或相减；求表面积时，只考虑外表面，不要重复计算被共用的内部表面。
For example, a solid consists of a cylinder of radius r and height h with a hemisphere of the same radius on top. The total volume is:
例如，一个立体由一个半径为r、高为h的圆柱和顶部同半径的半球组成。总体积为：
V = πr²h + (2/3)πr³
The total surface area is the cylinder’s curved area, the cylinder’s base and the hemisphere’s curved area:
总表面积为圆柱侧面积、圆柱底面和半球曲面积之和：
A = 2πrh + πr² + 2πr² = 2πrh + 3πr²
8. 3D Coordinates and Distance | 三维坐标与距离
In 3D coordinate geometry, a point P is represented by (x, y, z). The distance between two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) is given by the 3D distance formula:
在三维坐标几何中，点P用坐标(x, y, z)表示。两点P₁(x₁, y₁, z₁)和P₂(x₂, y₂, z₂)之间的距离由三维距离公式给出：
d = √((x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²)
The midpoint of the segment joining P₁ and P₂ is:
连接P₁和P₂的线段中点为：
( (x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2 )
These tools are useful for finding lengths of edges, diagonals and positions of points in 3D solids.
这些工具常用于求棱长、体对角线长度以及三维立体中点的位置。
9. Cross-sections and Projections | 截面与投影
A cross-section is the 2D shape formed by slicing a 3D solid with a plane. For a sphere, every cross-section is a circle. For a cone, a horizontal slice parallel to the base is a circle, while a vertical slice through the apex is a triangle. For a cylinder, a horizontal slice is a circle and an axial slice is a rectangle.
截面是用一个平面切割三维立体得到的二维形状。球的任何截面都是圆；圆锥平行于底面的水平截面是圆，通过顶点的竖直截面是三角形；圆柱的水平截面是圆，轴截面
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