📚 Mensuration: Volume & Surface Area | 测量法：体积与表面积
In IGCSE Maths, mensuration involves calculating lengths, areas, and volumes of 2D and 3D shapes. This article focuses on volume and surface area of prisms, cylinders, pyramids, cones, and spheres. You also need to work with compound shapes and convert between units such as cm³, litres, and m³.
在 IGCSE 数学中，测量法涉及计算二维和三维图形的长度、面积和体积。本文重点讲解棱柱、圆柱、棱锥、圆锥和球体的体积与表面积。你还需要掌握组合图形以及 cm³、升和 m³ 等单位之间的换算。
1. Key formulae at a glance | 核心公式速览
You must be confident with the standard results for volume and surface area. The exam can ask you to use them directly, rearrange them, or combine them for compound solids.
你必须熟练掌握体积和表面积的标准公式。考试可能会要求你直接使用、变形或者为组合体综合运用这些公式。
| Shape | Volume | Surface area |
|---|---|---|
| Prism | area of cross-section × length | sum of all exposed faces |
| Cylinder radius r, height h | πr²h | 2πr² + 2πrh |
| Pyramid | (1/3) × base area × h | sum of base and triangular faces |
| Cone radius r, height h, slant l | (1/3)πr²h | πr² + πrl |
| Sphere radius r | (4/3)πr³ | 4πr² |
Learn these formulae actively: cover the table and test yourself on both the symbol form and the word form.
要积极记忆这些公式：遮盖表格，测试自己能否写出符号形式和文字形式。
2. Prisms: volume | 棱柱：体积
A prism has a constant cross-section. Its volume is found by multiplying the area of the cross-section by the length of the prism.
棱柱具有恒定的横截面。其体积等于横截面面积乘以棱柱的长度。
V = area of cross-section × length
endicular height 5 cm, and the prism length is 12 cm, then V = (½ × 8 × 5) × 12 = 240 cm³.
对于三棱柱，先求三角形端面的面积。如果三角形底为 8 cm、垂直高为 5 cm，棱柱长度为 12 cm，则 V = (½ × 8 × 5) × 12 = 240 cm³。
Always use the perpendicular height of the cross-section, not a slant edge. The cross-section can be any polygon, such as a rectangle, triangle, trapezium, or a combination of shapes.
一定要使用横截面的垂直高，而不是斜边。横截面可以是任何多边形，例如矩形、三角形、梯形或者由多个图形组合而成。
3. Cylinders: volume and surface area | 圆柱：体积与表面积
m with a circular cross-section of radius r and height h. Its volume is given by V = πr²h.
圆柱是一种特殊的棱柱，横截面为半径为 r 的圆，高为 h。其体积为 V = πr²h。
V = πr²h
The total surface area of a closed cylinder is the sum of the two circular ends and the curved surface. The two ends contribute 2πr², and the curved surface area is 2πrh.
封闭圆柱的总表面积是两个圆形底面与弯曲侧面的面积之和。两个底面贡献 2πr²，弯曲侧面面积为 2πrh。
A = 2πr² + 2πrh
+ 2πrh. Leave your answer in terms of π unless the question gives a decimal approximation.
如果圆柱一端开口，则使用 A = πr² + 2πrh。除非题目要求给出小数近似，否则答案可保留 π。
4. Pyramids: volume | 棱锥：体积
A pyramid has a polygon base and triangular faces meeting at an apex. Its volume is one third of the volume of the surrounding prism with the same base and height.
棱锥有一个多边形底面，侧面为三角形并汇聚于顶点。其体积是与它同底等高的外接棱柱体积的三分之一。
V = (1/3) × base area × perpendicular height
are pyramid with base side 6 cm and vertical height 9 cm has V = (1/3) × 36 × 9 = 108 cm³.
要使用从顶点到底面的垂直高，而不是三角形侧面的斜高。例如，一个底面边长为 6 cm、垂直高为 9 cm 的正四棱锥，其体积为 V = (1/3) × 36 × 9 = 108 cm³。
If the base is a rectangle, triangle, or compound polygon, first calculate the base area, then multiply by one third of the height.
如果底面是矩形、三角形或组合多边形，先计算底面积，再乘以高度的三分之一。
5. Cones: volume and surface area | 圆锥：体积与表面积
A cone is a pyramid with a circular base. Its volume is one third of the volume of a cylinder with the same radius and height.
圆锥是底面为圆的棱锥。其体积是与它同半径、同高的圆柱体积的三分之一。
V = (1/3)πr²h
For total surface area, add the base circle πr² to the curved surface area πrl, where l is the slant height.
总表面积为底面圆面积 πr² 加上弯曲侧面面积 πrl，其中 l 为斜高。
A = πr² + πrl
If you are given the radius r and the perpendicular height h, find the slant height using Pythagoras' theorem.
如果给出半径 r 和垂直高 h，使用勾股定理求斜高。
l² = r² + h²
Do not confuse l and h in the formula. Volume uses h, but the curved surface area uses l.
不要混淆公式中的 l 和 h。体积使用 h，但弯曲侧面面积使用 l。
6. Spheres and hemispheres | 球体与半球体
These two formulae are given in the IGCSE exam but you should be able to use them quickly.
半径为 r 的球体体积为 V = (4/3)πr³，表面积为 A = 4πr²。这两个公式在 IGCSE 考试中通常给出，但你应能快速使用。
V = (4/3)πr³, A = 4πr²
rface area includes the curved surface 2πr² plus the flat circular base πr², giving a total of 3πr².
半球体的体积是球体体积的一半：V = (2/3)πr³。其表面积包括弯曲曲面面积 2πr² 加上平面圆形底面 πr²，总表面积为 3πr²。
V = (2/3)πr³, A = 3πr²
Many candidates forget the flat circular face when finding the surface area of a hemisphere. Include it unless the question asks for the curved surface area only.
许多考生在求半球体表面积时会忘记平面圆形底面。除非题目只要求弯曲曲面面积，否则一定要加上它。
7. Surface area of 3D solids | 三维图形的表面积
To find surface area, draw a net or list all exposed faces. Add the area of each exposed face. Do not include internal faces or faces hidden inside compound solids.
求表面积时，画出展开图或列出所有外露面。将每个外露面的面积相加。不要计算内部面或组合体中被隐藏的面。
For a cuboid of length l, width w, and height h, the total surface area is 2lw + 2lh + 2wh. This comes from the three pairs of opposite faces.
对于长 l、宽 w、高 h 的长方体，总表面积为 2lw + 2lh + 2wh。这是由三对相对的面得到的。
For a closed prism, include the two cross-section faces and all rectangular side faces. Always give the final surface area in square units such as cm² or m².
对于封闭棱柱，要包括两个横截面面和所有矩形侧面。最终表面积一定要使用平方单位，如 cm² 或 m²。
8. Compound shapes | 组合图形
For compound solids, split the shape into simple solids, calculate each volume or area, and then add or subtract as needed.
对于组合体，将图形拆分为简单体，分别计算体积或表面积，然后根据需要相加或相减。
For example, a grain container is made from a cylinder with a hemisphere on top. The total volume is the volume of the cylinder plus the volume of the hemisphere. The total surface area is the curved surface of the cylinder plus the surface area of the hemisphere, but the circular base where they meet is not exposed.
例如，一个谷物容器由一个圆柱和顶部一个半球组成。总体积等于圆柱体积加半球体积。总表面积等于圆柱的弯曲侧面面积加半球表面积，但两者相接的圆面不外露，不计入。
Always identify the shared faces when adding or subtracting surface areas of compound solids. These shared faces are internal and should not be counted.
在加或减组合体的表面积时，一定要识别共用面。这些共用面属于内部面，不应计算。
9. Units and conversions | 单位与换算
m³. Capacity units are related to volume: 1 cm³ = 1 mL, 1000 cm³ = 1 L, and 1 m³ = 1000 L.
体积单位为立方单位：cm³、m³、mm³。容量单位与体积相关：1 cm³ = 1 mL，1000 cm³ = 1 L，1 m³ = 1000 L。
verting squared or cubed units: multiply or divide by the factor raised to the power of the dimension.
面积中，1 m² = 10000 cm²。体积中，1 m³ = 1000000 cm³。换算平方或立方单位时要小心：要乘以或除以相应倍数的平方或立方。
If a question gives measurements in different units, convert them all to the same unit before calculating. For example, convert 2 m to 200 cm before using it with other centimetre measurements.
如果题目给出的测量值单位不同，计算前要全部换算成相同单位
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