📚 Volume and Surface Area: Mastering 3D Mensuration for IGCSE | 体积与表面积：攻克IGCSE三维度量
In the IGCSE Mathematics extended syllabus (0580), mensuration of three-dimensional shapes appears in every examination session. You may be asked to find the volume of a prism, the surface area of a cylinder, or the capacity of a composite container. A clear command of formulas, units, and problem-solving structure is essential to score full marks.
在IGCSE数学扩展课程（0580）中，三维图形的度量在每场考试中都会出现。你可能会被要求计算棱柱的体积、圆柱的表面积或复合容器的容量。熟练掌握公式、单位换算和解题步骤是获得满分的关键。
1. Know Your 3D Shapes | 认识你的立体图形
Before applying any formula, you must confidently identify the shape in the question. A prism has a uniform cross-section, a cylinder is a circular prism, a cone and pyramid taper to a point, and a sphere is perfectly round. The key dimensions to identify are the radius r, the perpendicular height h, the slant height l, and the area of the base or cross-section.
在套用任何公式之前，你必须能准确辨认题目中的立体图形。棱柱具有均匀的横截面，圆柱是圆形棱柱，圆锥和棱锥向顶点收拢，球体则完美对称。关键尺寸包括半径r、垂直高h、斜高l以及底面或横截面的面积。
| Shape 图形 | Key dimensions 关键尺寸 | Volume formula 体积公式 |
| Cuboid 长方体 | length l, width w, height h | l × w × h |
| Cylinder 圆柱 | radius r, height h | πr²h |
| Cone 圆锥 | radius r, height h, slant l | ⅓πr²h |
| Sphere 球体 | radius r | (4/3)πr³ |
2. Volume of Prisms | 棱柱的体积
For any prism, the volume is found by multiplying the area of the cross-section by the length (or height) of the prism. The most common prisms in IGCSE are the cuboid, the triangular prism, and the cylinder.
对任何棱柱，体积等于横截面积乘以棱柱的长度（或高度）。IGCSE中最常见的棱柱是长方体、三棱柱和圆柱。
V = area of cross-section × length
Worked example: a triangular prism has a cross-section with base 4 cm and height 3 cm, and its length is 10 cm. The area of the cross-section is ½ × 4 × 3 = 6 cm², so the volume is 6 × 10 = 60 cm³.
例题：一个三棱柱的横截面底为4 cm、高为3 cm，棱柱长度为10 cm。横截面积为½×4×3=6 cm²，因此体积为6×10=60 cm³。
Remember that the cross-section must be the face that is identical all the way through the prism. In a triangular prism, use the triangle area formula; in a trapezoidal prism, use the trapezium area formula.
请记住，横截面必须是沿棱柱贯穿始终的相同截面。在三棱柱中使用三角形面积公式，在梯形棱柱中使用梯形面积公式。
3. Volume of a Cylinder | 圆柱的体积
A cylinder is a prism with a circular cross-section, so its volume is the area of a circle multiplied by the height of the cylinder. This formula is given on the IGCSE formula sheet but you must know exactly how to use it.
圆柱是圆形横截面的棱柱，因此其体积等于圆面积乘以圆柱高度。该公式印在IGCSE公式表中，但你必须精确掌握其用法。
V = πr²h
Worked example: a cylinder has radius 5 cm and height 12 cm. Its volume is V = π × 5² × 12 = 300π ≈ 942.5 cm³. Always give both the
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