Mensuration: Volume and Surface Area of Prisms and Cylinders | IGCSE 数学：棱柱与圆柱的体积和表面积

📚 Mensuration: Volume and Surface Area of Prisms and Cylinders | IGCSE 数学：棱柱与圆柱的体积和表面积

Mensuration is the branch of IGCSE Mathematics that deals with measuring lengths, areas, and volumes of 2D and 3D shapes. In this revision article, we focus on prisms and cylinders: how to calculate their volume, how to find surface area, and how to solve problems involving composite solids and unit conversion.

测算是 IGCSE 数学中研究二维和三维图形长度、面积和体积度量的一部分。本文重点复习棱柱与圆柱：如何计算它们的体积、如何求表面积，以及如何解决涉及组合体和单位换算的问题。

1. What Is a Prism? | 什么是棱柱？

A prism is a 3D solid with a constant cross-section. This means that if you slice the solid parallel to its end face, every slice has exactly the same shape and size. Common examples include cuboids, triangular prisms, and hexagonal prisms.

棱柱是一种具有恒定横截面的三维立体。也就是说，如果平行于端面切割该立体，每一片的形状和大小完全相同。常见的例子包括长方体、三棱柱和六棱柱。

A cylinder is similar to a prism because it also has a constant cross-section, but its cross-section is a circle rather than a polygon. In IGCSE questions, the same principle is used: volume equals area of cross-section times length.

圆柱与棱柱类似，因为它也具有恒定横截面，但横截面是圆而不是多边形。在 IGCSE 题目中，使用的原理相同：体积等于横截面积乘以长度。

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2. Volume of a Prism | 棱柱的体积

The volume of any prism is found by multiplying the area of its cross-section by the length between the two parallel end faces. The general formula is:

任何棱柱的体积都等于其横截面积乘以两个平行端面之间的长度。通用公式为：

V = A × l

Here V is volume, A is the cross-sectional area, and l is the length of the prism. For a cuboid with length l, width w, and height h, the cross-sectional area is A = l × w, so the volume becomes V = l × w × h.

其中 V 是体积，A 是横截面积，l 是棱柱的长度。对于长为 l、宽为 w、高为 h 的长方体，横截面积为 A = l × w，因此体积为 V = l × w × h。

For a triangular prism, first calculate the area of the triangular end using A = ½ × base × height, then multiply by the length of the prism. Do not confuse the height of the triangle with the length of the prism.

对于三棱柱，首先使用 A = ½ × 底 × 高 计算三角形端面的面积，然后乘以棱柱的长度。不要将三角形的高与棱柱的长度混淆。

• Volume of a prism = area of cross-section × length

  棱柱体积 = 横截面积 × 长度

• Volume of a triangular prism = ½ × b × h_triangle × l

  三棱柱体积 = ½ × b × h三角形 × l

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3. Surface Area of a Prism | 棱柱的表面积

Surface area is the total area of all faces of a 3D solid. To find the surface area of a prism, calculate the area of each face separately and then add them together. This is safer than trying to memorise a single formula because prism shapes vary.

表面积是三维立体所有面的总面积。求棱柱表面积时，应分别计算每个面的面积，然后相加。这比试图记住单一公式更稳妥，因为棱柱形状多种多样。

For a closed triangular prism, there are two triangular ends and three rectangular side faces. For a cuboid, there are six rectangular faces. In exam questions, you may need to find missing lengths using Pythagoras’ theorem before calculating area.

对于封闭的三棱柱，有两个三角形端面和三个矩形侧面。对于长方体，有六个矩形面。在考试题中，你可能需要先用勾股定理求出缺失的长度，再计算面积。

A useful check is to visualise the net of the solid. Opening the shape into a flat net helps you count every face and avoid missing one.

一个有用的检查方法是想象立体的展开图。将图形展开成平面展开图有助于你数清每个面，避免遗漏。

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4. Volume of a Cylinder | 圆柱的体积

A cylinder has a circular cross-section. The area of the circular end is A = πr², where r is the radius of the circle. Therefore, the volume of a cylinder is given by:

圆柱具有圆形横截面。圆形端面的面积为 A = πr²，其中 r 是圆的半径。因此，圆柱体积由下式给出：

V = πr²h

Here h is the height or length of the cylinder. You must use the radius, not the diameter. If a question gives the diameter, divide it by 2 first.

其中 h 是圆柱的高或长度。必须使用半径，而不是直径。如果题目给出直径，请先除以 2。

When the answer is required to a suitable degree of accuracy, use the π button on your calculator rather than 3.14 unless the question specifies an approximation. If no rounding instruction is given, give an answer in terms of π.

当答案要求合适的精确度时，请使用计算器上的 π 按键，而不是 3.14，除非题目明确指定近似值。如果没有给出舍入要求，请以 π 的形式给出答案。

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5. Surface Area of a Cylinder | 圆柱的表面积

The surface area of a closed cylinder consists of two circles and one curved rectangular surface. The two circular ends each have area πr², and the curved surface has area 2πrh. Therefore:

封闭圆柱的表面积由两个圆和一个曲面矩形组成。两个圆形端面每个面积为 πr²，曲面面积为 2πrh。因此：

A = 2πr² + 2πrh

This formula is given in the IGCSE formula sheet, but you must be able to choose it correctly. The term 2πr² covers the top and bottom circles, while 2πrh is the curved surface area.

该公式在 IGCSE 公式表中有给出，但你必须能够正确选择使用。项 2πr² 覆盖上底和下底两个圆，而 2πrh 是曲面面积。

If the cylinder is open at one end, use A = πr² + 2πrh. If it is a hollow tube open at both ends, use A = 2πrh only. Read the question carefully to see whether the solid is open or closed.

如果圆柱一端开口，则使用 A = πr² + 2πrh。如果它是两端开口的空心管，则只使用 A = 2πrh。仔细阅读题目，判断该立体是开口还是封闭。

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6. Units and Conversions | 单位与换算

Length is measured in mm, cm, m, and km. Area is measured in square units such as cm², m², and mm². Volume is measured in cubic units such as cm³, m³, and litres. You must convert all measurements to the same unit before calculating.

长度以毫米、厘米、米和千米为单位。面积以平方单位计量，如 cm²、m² 和 mm²。体积以立方单位计量，如 cm³、m³ 和升。在计算之前，必须将所有测量值换算为相同单位。

Quantity	Conversion
Length	1 m = 100 cm, 1 cm = 10 mm
Area	1 m² = 10 000 cm², 1 cm² = 100 mm²
Volume	1 m³ = 1 000 000 cm³, 1 litre = 1000 cm³

A common mistake is to convert 1 m³ to 100 cm³. This is wrong because volume scales by the cube of the linear scale factor: 100 × 100 × 100 = 1 000 000.

一个常见错误是把 1 m³ 换算成 100 cm³。这是错误的，因为体积按线性比例因子的三次方变化：100 × 100 × 100 = 1 000 000。

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7. Composite Solids and Missing Lengths | 组合体与缺失长度

IGCSE questions often ask you to find the volume or surface area of a composite solid made from two or more simpler shapes. The key strategy is to split the solid

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