📚 Volume of Cuboids, Cubes and Composite Solids | 长方体、正方体与组合体的体积
In IGCSE Mathematics, volume measures the amount of three-dimensional space occupied by a solid object. This article focuses on finding the volume of cuboids (rectangular prisms), cubes, and composite solids constructed from these basic shapes.
在 IGCSE 数学中，体积测量的是立体物体所占三维空间的大小。本文重点讲解如何求长方体、正方体以及由这些基本图形组合而成的组合体的体积。
You need to be confident with using cubic units, converting between related units, and solving problems where a dimension is missing or where a shape must be split into simpler parts.
你需要熟练掌握立方单位的使用、相关单位之间的换算，并能解决缺少某个边长或需要将图形分割为简单部分的题目。
1. What Is Volume? | 什么是体积？
Volume is the amount of space inside a three-dimensional object. It is not the same as area, which measures a flat surface, nor the same as perimeter, which measures the distance around a shape.
体积是三维物体内部空间的大小。它不同于面积，面积测量的是平面图形的大小；也不同于周长，周长测量的是图形边界的长度。
In the IGCSE core course, the most common volume problems involve cuboids. A cuboid is a box-shaped solid with six rectangular faces. Every corner is a right angle.
在 IGCSE 核心课程中，最常见的体积问题都涉及长方体。长方体是由六个长方形面组成的盒子形状的立体，每个角都是直角。
We always record volume in cubic units, such as cubic centimetres (cm³), cubic metres (m³), cubic millimetres (mm³), and sometimes litres for liquids.
我们始终用立方单位记录体积，例如立方厘米（cm³）、立方米（m³）、立方毫米（mm³），有时也用升来表示液体体积。
2. Unit Cubes and Cubic Units | 单位立方体与立方单位
To understand the volume formula, imagine filling a cuboid with centimetre cubes. Each small cube has side length 1 cm and volume 1 cm³.
为了理解体积公式，可以想象用厘米立方体填满一个长方体。每个小立方体的边长为 1 cm，体积为 1 cm³。
If a cuboid has length 5 cm, width 3 cm, and height 2 cm, you can place 5 cubes along the length, 3 cubes along the width, and 2 layers high.
如果一个长方体的长为 5 cm、宽为 3 cm、高为 2 cm，那么你可以沿着长放 5 个立方体，沿着宽放 3 个立方体，高度方向有 2 层。
The total number of unit cubes is found by multiplying the three dimensions:
单位立方体的总数由三个维度相乘得到：
Number of unit cubes = length × width × height
单位立方体数量 = 长 × 宽 × 高
For the example above, the number of cubes is 5 × 3 × 2 = 30, so the volume is 30 cm³.
在上面的例子中，立方体数量为 5 × 3 × 2 = 30，因此体积为 30 cm³。
3. Volume of a Cuboid: The Formula | 长方体体积公式
For any cuboid, the volume is given by the formula:
对于任何长方体，体积公式为：
V = l × w × h
体积 = 长 × 宽 × 高
Here, l is the length, w is the width, and h is the height of the cuboid. The order does not matter because multiplication is commutative.
这里，l 表示长，w 表示宽，h 表示高。顺序并不重要，因为乘法满足交换律。
e = base area × height. The base area is l × w, so multiplying by h gives the same result.
该公式也可以写成体积 = 底面积 × 高。底面积为 l × w，再乘以 h 得到相同结果。
uboid has length 8 cm, width 5 cm, and height 4 cm. Its volume is V = 8 × 5 × 4 = 160 cm³.
例题：一个长方体的长为 8 cm、宽为 5 cm、高为 4 cm。它的体积为 V = 8 × 5 × 4 = 160 cm³。
4. Volume of a Cube | 正方体体积
A cube is a special cuboid where all three dimensions are equal. If the side length is a, then the volume formula becomes:
正方体是一种特殊的长方体，它的三个维度都相等。如果边长为 a，则体积公式变为：
V = a × a × a = a³
体积 = a × a × a = a³
de length cubed. For example, a cube with side length 6 cm has volume 6 × 6 × 6 = 216 cm³.
这意味着正方体的体积等于边长的三次方。例如，边长为 6 cm 的正方体体积为 6 × 6 × 6 = 216 cm³。
IGCSE questions often give the volume of a cube and ask you to find the side length. In that case, take the cube root of the volume.
IGCSE 考题经常给出正方体的体积，要求你求边长。这时需要将体积开三次方。
cm³, then its side length is the cube root of 125, which is 5 cm, because 5 × 5 × 5 = 125.
例题：如果正方体的体积为 125 cm³，那么它的边长为 125 的立方根，即 5 cm，因为 5 × 5 × 5 = 125。
5. Working with Mixed Units | 混合单位的换算
Volume calculations become tricky when lengths are given in different units. You must convert all dimensions to the same unit before multiplying.
当长度使用不同单位时，体积计算会变得容易出错。你必须先将所有维度转换为相同单位，再进行乘法计算。
The most important conversions are:
最重要的换算关系有：
| Length | Volume |
| 1 m = 100 cm | 1 m³ = 1,000,000 cm³ |
| 1 cm = 10 mm | 1 cm³ = 1,000 mm³ |
| 1 L = 1,000 cm³ | 1 mL = 1 cm³ |
engths to cm: 0.5 m = 50 cm, 300 mm = 30 cm. Then the volume is 50 × 20 × 30 = 30,000 cm³.
例如，一个长方体的尺寸为 0.5 m × 20 cm × 300 mm。首先将所有长度转换为 cm：0.5 m = 50 cm，300 mm = 30 cm。然后体积为 50 × 20 × 30 = 30,000 cm³。
Never multiply numbers with mixed units without converting, because the result will not be a meaningful cubic unit.
切勿在未换算的情况下将混合单位直接相乘，因为得到的结果将不是有意义的立方单位。
6. Finding a Missing Dimension | 求缺失的边长
Some problems give the volume and ask you to find one missing length. Rearrange the cuboid formula to make the unknown dimension the subject.
有些题目给出体积并要求你求某个缺失的长度。此时需要将长方体公式变形，使未知维度成为主项。
If V = l × w × h, then:
如果 V = l × w × h，那么：
h = V ÷ (l × w)
高 = 体积 ÷（长 × 宽）
Example: A cuboid has volume 240 cm³, length 10 cm, and width 4 cm. Find its height.
例题：一个长方体体积为 240 cm³，长为 10 cm，宽为 4 cm。求它的高。
First compute l × w = 10 × 4 = 40 cm². Then h = 240 ÷ 40 = 6 cm.
首先计算 l × w = 10 × 4 = 40 cm²。然后 h = 240 ÷ 40 = 6 cm。
The same method works for finding length or width. Divide the volume by the product of the two known dimensions.
同样的方法也适用于求长或宽。将体积除以两个已知维度的乘积即可。
7. Composite Figures: Split and Add | 组合体：先分割再相加
A composite solid is made from two or more cuboids joined together. To find its total volume, split the shape into smaller cuboids, find each volume, and then add them.
组合体是由两个或多个长方体连接而成的立体。求它的总体积时，需要将图形分割成若干个长方体，分别求出体积，然后相加。
8 cm × 3 cm × 2 cm attached to a bottom cuboid 5 cm × 4 cm × 3 cm. Find the total volume.
例题：一个 L 形组合体由一个 8 cm × 3 cm × 2 cm 的上方长方体连接一个 5 cm × 4 cm × 3 cm 的下方长方体组成。求总体积。
2 = 48 cm³. Volume of bottom cuboid: 5 × 4 × 3 = 60 cm³. Total volume = 48 + 60 = 108 cm³.
上方长方体体积：8 × 3 × 2 = 48 cm³。下方长方体体积
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