Volume and Surface Area of 3D Shapes | 三维图形的体积与表面积

📚 Volume and Surface Area of 3D Shapes | 三维图形的体积与表面积

In IGCSE Mathematics, mensuration questions on 3D shapes ask you to calculate volume, curved surface area, total surface area and sometimes a missing length. You must be able to identify the shape, select the correct formula, use perpendicular height where required, and give the correct units.

在 IGCSE 数学中，三维图形的测量题要求你计算体积、侧面积、总表面积，有时还要求计算缺失的边长。你必须能够识别图形、选择正确的公式、在需要时使用垂直高度，并给出正确的单位。

This revision guide covers prisms, cylinders, pyramids, cones, spheres, hemispheres, composite solids and scale factors. It also highlights the mistakes that most often cost marks in school exams and IGCSE papers.

本复习指南涵盖棱柱、圆柱、棱锥、圆锥、球体、半球、组合体以及比例因子。同时也会指出学校考试和 IGCSE 试卷中最容易失分的常见错误。

1. Dimensions and Units | 维度与单位

Length is one-dimensional, area is two-dimensional and volume is three-dimensional. When you convert between units, remember that a length factor is squared for area and cubed for volume. This is why unit conversion errors are so common in mensuration.

长度是一维的，面积是二维的，体积是三维的。在进行单位换算时，要记住面积要乘以长度因子的平方，体积要乘以长度因子的立方。这就是测量题中单位换算错误非常常见的原因。

For example, 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. Capacity units are also important: 1 litre = 1000 cm³ and 1 ml = 1 cm³.

例如，1 m = 100 cm，因此 1 m² = 10,000 cm²，1 m³ = 1,000,000 cm³。容量单位也很重要：1 升 = 1000 cm³，1 毫升 = 1 cm³。

1 litre = 1000 cm³

2. Prisms: Volume and Surface Area | 棱柱的体积与表面积

A prism is a 3D shape with a constant cross-section. The volume of any prism is the area of the cross-section multiplied by the length. This rule works for cuboids, triangular prisms, trapezoidal prisms and any other prism with parallel identical ends.

棱柱是一种截面保持不变的立体图形。任何棱柱的体积都等于截面面积乘以长度。这个规则适用于长方体、三棱柱、梯形棱柱以及任何两端平行且形状相同的棱柱。

The total surface area of a prism is made up of two identical cross-section faces plus the side faces. For a cuboid with length l, width w and height h, the volume is V = lwh and the surface area is SA = 2lw + 2lh + 2wh.

棱柱的总表面积由两个相同的截面加上侧面面积组成。对于长为 l、宽为 w、高为 h 的长方体，体积为 V = lwh，表面积为 SA = 2lw + 2lh + 2wh。

For a triangular prism, you often need to find the area of the triangular face first. The height used for the triangular area must be perpendicular to the base of the triangle, not the length of the prism.

对于三棱柱，你通常需要先求出三角形面的面积。用于三角形面积的高必须垂直于三角形的底边，而不是棱柱的长度。

V = area of cross-section × length

SA = 2 × base area + perimeter × length

3. Cylinders | 圆柱

A cylinder is a special prism with a circular cross-section. Its volume is the area of the circular base multiplied by the perpendicular height. If the radius is r and the height is h, then V = πr²h.

圆柱是一种截面为圆形的特殊棱柱。它的体积等于圆底面积乘以垂直高度。如果底面半径为 r，高度为 h，则 V = πr²h。

The curved surface area of a cylinder is the area of the rectangle formed when the curved surface is opened out. It is given by CSA = 2πrh. The total surface area adds the two circular ends: SA = 2πrh + 2πr².

圆柱的侧面积是曲面展开后形成的矩形面积。公式为 CSA = 2πrh。总表面积还要加上两个圆形底面：SA = 2πrh + 2πr²。

If the question gives the diameter, divide by 2 to find the radius before substituting into the formula. If the question asks for an answer in terms of π, leave the multiple of π rather than using 3.14.

如果题目给出直径，先除以 2 求出半径，再代入公式。如果题目要求答案用 π 表示，就保留 π 的倍数，而不要用 3.14 代替。

V = πr²h

CSA = 2πrh SA = 2πrh + 2πr²

4. Pyramids and Cones | 棱锥与圆锥

A pyramid has a polygon base and triangular faces meeting at an apex. Its volume

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