Converting between Units of Volume | 体积单位换算

📚 Converting between Units of Volume | 体积单位换算

Volume measures the amount of space a three-dimensional object occupies. In the Edexcel IGCSE Mathematics syllabus, you are expected to convert fluently between units such as millilitres, litres, cubic centimetres and cubic metres, and to apply these conversions in problems involving prisms, cylinders and density.

体积衡量三维物体所占空间的大小。在爱德思 IGCSE 数学考纲中，学生需要熟练地在毫升、升、立方厘米和立方米等单位之间进行换算，并在棱柱、圆柱及密度相关问题中加以应用。

1. The Metric System of Volume | 公制体积单位体系

The metric system is built on powers of ten. The scientific base unit of volume is the cubic metre (m³), but in everyday life we often use the litre (L).

公制单位以十的幂次为基础。科学中体积的基本单位是立方米（m³），而日常生活中我们常使用升（L）。

Cubic millimetre: mm³, used for very small volumes such as a drop of ink.

立方毫米（mm³）：用于非常小的体积，例如一滴墨水。
Cubic centimetre: cm³, commonly used in science experiments.

立方厘米（cm³）：常用于科学实验。
Cubic metre: m³, used for rooms, pools and large containers.

立方米（m³）：用于房间、泳池和大型容器。
Litre (L) and millilitre (mL), used for liquids and food packaging.

升（L）和毫升（mL）：用于液体和食品包装。

Each step on the length scale changes the volume scale by the cube of the length factor. Understanding this single rule unlocks every conversion in this chapter.

长度尺度每变化一步，体积尺度就要按长度倍数的三次方变化。理解这一条规则，就能解开本章所有换算。

2. The Cubic Relationship | 立方换算关系

If a length is measured in metres and we multiply by 100 to obtain centimetres, then a volume measured in m³ must be multiplied by 100³ = 1,000,000 to obtain cm³.

如果长度以米为单位，乘以 100 可换算为厘米，那么以立方米为单位的体积要乘以 100³ = 1,000,000 才能换算为立方厘米。

1 m³ = (100 cm)³ = 100³ cm³ = 1,000,000 cm³

Similarly, since 1 cm = 10 mm, applying the cube gives:

同理，由于 1 cm = 10 mm，取三次方可得：

1 cm³ = (10 mm)³ = 10³ mm³ = 1,000 mm³

This is the single most important idea in this topic: cube the length conversion factor. Do not just multiply or divide by 100; you must handle the exponent correctly.

这就是本专题最重要的思路：将长度换算系数取三次方。不要只乘以或除以 100，必须正确处理三次方指数。

3. Converting Between Cubic Units | 立方单位之间互化

To convert from a smaller unit to a larger unit, divide by the conversion factor. To convert from a larger unit to a smaller unit, multiply by the conversion factor.

从小单位化到大单位，除以换算系数；从大单位化到小单位，乘以换算系数。

Example 1: Convert 5.2 m³ to cm³.

例1：将 5.2 m³ 换算为 cm³。

Since m³ is the larger unit and cm³ is smaller, we multiply.

因为 m³ 是大单位，cm³ 是小单位，所以用乘法。

5.2 × 1,000,000 = 5,200,000 cm³

Example 2: Convert 3,400 mm³ to cm³.

例2：将 3,400 mm³ 换算为 cm³。

Here we move from a smaller unit to a larger one, so we divide.

这里从小单位化到大单位，所以用除法。

3,400 ÷ 1,000 = 3.4 cm³

Always ask yourself: is the new unit larger or smaller? Your numeric answer should grow or shrink accordingly.

做题时始终问自己：新单位更大还是更小？数值应相应增大或缩小。

4. Litres and Millilitres | 升与毫升

The litre is defined such that 1 L = 1,000 mL. Crucially for scientific work, 1 mL = 1 cm³.

升的定义为 1 L = 1,000 mL。对科学计算尤为关键的是：1 mL = 1 cm³。

1 L = 1,000 cm³

This means that a cube with side length 10 cm has a volume of exactly one litre:

这意味着一个棱长为 10 cm 的立方体，其体积恰好为 1 升：

10 cm × 10 cm × 10 cm = 1,000 cm³ = 1 L

This link between the cubic scale and the liquid scale is tested frequently in IGCSE questions, especially in contexts such as measuring jugs, bottles and medicine doses.

体积单位与液体单位之间的联系，是 IGCSE 高频考点，尤其常见于量杯、水瓶和药剂剂量等情境。

5. Cubic Metres and Litres | 立方米与升

Since 1 m³ = 1,000,000 cm³ and 1 L = 1,000 cm³, we divide to connect the two:

因为 1 m³ = 1,000,000 cm³ 且 1 L = 1,000 cm³，相除即可建立联系：

1 m³ = 1,000 L

A water tank of volume 2.5 m³ therefore holds 2,500 litres of water.

因此，体积为 2.5 m³ 的水箱可容纳 2,500 升水。

1 m³ = 1,000 L

1 立方米 = 1,000 升
1 L = 0.001 m³

1 升 = 0.001 立方米
1 L = 1,000 mL = 1,000 cm³

1 升 = 1,000 毫升 = 1,000 立方厘米

Memorise these three facts and you can navigate every volume conversion in this chapter with confidence.

记住这三个事实，你就能自信地完成本章所有体积换算。

6. Working with Mixed Units | 混合单位运算

When a question gives measurements in different units, convert everything to the same unit before calculating. This is the most common source of lost marks on IGCSE paper questions.

当题目给出不同单位的测量值时，先统一单位再计算。这是 IGCSE 试卷上最常见的失分点。

Example 3: A cuboid measures 0.5 m × 20 cm × 300 mm. Find its volume in cm³.

例3：一个长方体长 0.5 m、宽 20 cm、高 300 mm，求体积（以 cm³ 为单位）。

Convert each dimension to centimetres before multiplying:

先将各边统一换算为厘米再相乘：

0.5 m = 50 cm, 20 cm = 20 cm, 300 mm = 30 cm

V = 50 × 20 × 30 = 30,000 cm³

Always write the unit next to every number in your working. This simple habit prevents mixing errors and helps examiners see your method clearly.

在每一步计算中都要写上单位。这个简单习惯能防止混用单位的错误，也能让阅卷老师清楚看到你的思路。

7. Volume of Prisms and Cylinders | 棱柱与圆柱的体积

For any prism, volume = area of cross-section × length. For a cylinder, volume = πr²h. Unit conversion often appears inside these formulas.

对任何棱柱，体积 = 底面积 × 高。对圆柱，体积 = πr²h。单位换算常出现在这些公式的实际计算中。

Example 4: A cylinder has radius 7 cm and height 0.12 m. Calculate its volume in cm³.

例4：一个圆柱底面半径 7 cm，高 0.12 m，求体积（以 cm³ 为单位）。

First convert the height to centimetres:

先将高换算为厘米：

0.12 m = 12 cm

V = π × 7² × 12 = π × 49 × 12 = 588π ≈ 1,847 cm³

Notice how converting the height early simplified the rest of the calculation. If you convert after substitution, you must cube the length conversion, which is far more error-prone.

注意，提前换算高度使后续计算变得简单。如果代入后再换算，就需要对长度系数取三次方，更容易出错。

8. Density and Unit Conversion | 密度与单位换算

The density formula ρ = m ÷ V links mass and volume. Density problems often require you to convert volume units before dividing.

密度公式 ρ = m ÷ V 将质量与体积联系起来。密度题通常需要先换算体积单位再作除法。

Example 5: A substance has mass 250 g and volume 400 cm³. Find its density in g/cm³.

例5：某物质质量为 250 g，体积为 400 cm³，求密度（以 g/cm³ 为单位）。

ρ = 250 ÷ 400 = 0.625 g/cm³

If the volume had been given in m³, you would first convert it to cm³, or alternatively express mass in kg to work in kg/m³. The key principle is to keep mass and volume units compatible with each other.

若体积以 m³ 给出，则应先换算为 cm³，或者用 kg 表示质量从而使用 kg/m³。关键原则是保持质量与体积单位相互匹配。

9. Common Exam Pitfalls | 常见考试陷阱

Forgetting to cube the conversion factor: 1 m = 100 cm, but 1 m³ = 1,000,000 cm³, not 100 cm³.

忘记对换算系数取三次方：1 m = 100 cm，但 1 m³ = 1,000,000 cm³，而非 100 cm³。
Confusing millilitres with cubic metres: write the units in order from smallest to largest before converting, e.g. mm³ → cm³ → m³.

混淆毫升与立方米：换算前按从小到大的顺序排列单位，例如 mm³ → cm³ → m³。
Using the wrong operation: when moving to a larger unit, the number must decrease; when moving to a smaller unit, it must increase.

运算方向用反：化为更大单位时数值必须变小；化为更小单位时数值必须变大。
Omitting units in the final answer: IGCSE mark schemes require the correct unit to earn full marks.

最后答案漏写单位：IGCSE 评分标准要求写出正确单位才能得满分。

Avoiding these four errors will immediately improve your accuracy on volume conversion questions.Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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Converting between Units of Volume | 体积单位换算