Volume of Solid Figures | 立体图形的体积

📚 Volume of Solid Figures | 立体图形的体积

Volume is the amount of space a solid figure occupies. We measure volume in cubic units, such as cubic centimetres (cm³) or cubic metres (m³), because we are filling space in three directions: length, width, and height.

体积是指一个立体图形所占空间的大小。我们用立方单位来测量体积，例如立方厘米（cm³）或立方米（m³），因为我们需要从长、宽、高三个方向来理解所占据的空间。

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1. What Is Volume? | 什么是体积？

A solid figure has three dimensions. When we talk about volume, we are asking: ‘How many unit cubes can fit inside this shape?’ A unit cube is a cube that measures 1 unit on every side, so its volume is 1 cubic unit.

立体图形有三个维度。当我们讨论体积时，实际上是在问：“这个形状里面可以放多少个单位立方体？”单位立方体是指每条边长度都为1单位的立方体，因此它的体积是1立方单位。

For example, a small box that can hold exactly 8 unit cubes has a volume of 8 cubic units. The number of unit cubes completely filling the space tells us the volume.

例如，一个小盒子恰好能装下8个单位立方体，那么它的体积就是8立方单位。完全填满这个空间的单位立方体的数量，就是我们所说的体积。

• Volume is measured in cubic units: cm³, m³, in³, ft³.
• 体积用立方单位来测量：cm³，m³，in³，ft³。
• A unit cube has side length 1 unit and volume 1 cubic unit.
• 一个单位立方体的边长是1单位，体积是1立方单位。

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2. Units of Volume | 体积的单位

Because volume involves three dimensions, we use ‘cubic’ units. If the side length is measured in centimetres, the volume is in cubic centimetres (cm³). If the side length is measured in metres, the volume is in cubic metres (m³).

由于体积涉及三个维度，我们使用“立方”单位。如果边长以厘米为单位，体积就以立方厘米（cm³）为单位；如果边长以米为单位，体积就以立方米（m³）为单位。

Unit | 单位	Used for | 用途
cm³	Small boxes, cubes, containers | 小盒子、立方体、容器
m³	Rooms, large storage tanks | 房间、大型储水箱
in³ / ft³	Imperial units for small and medium solids | 英制单位，用于中小型立体

Remember: a square with side 2 cm has area 2 × 2 = 4 cm², but a cube with side 2 cm has volume 2 × 2 × 2 = 8 cm³. The ‘cubic’ part tells us we multiplied three lengths together.

请记住：边长为2厘米的正方形面积是2×2=4 cm²，但棱长为2厘米的立方体体积是2×2×2=8 cm³。“立方”一词表示我们将三个长度相乘。

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3. Counting Unit Cubes | 数单位立方体

One simple way to find volume is to count the unit cubes inside a solid figure. If the figure is fully packed with unit cubes, the volume is simply the total number of cubes.

求体积的一个简单方法是数立体图形中的单位立方体。如果这个图形完全由单位立方体铺满，那么体积就是这些立方体的总数。

Look at a rectangular prism that is 3 cubes long, 2 cubes wide, and 2 cubes high. We can count: the bottom layer has 3 × 2 = 6 cubes, and there are 2 layers. So the total is 3 × 2 × 2 = 12 cubes.

观察一个长3个立方体、宽2个立方体、高2个立方体的长方体。我们可以数一数：底层有3×2=6个立方体，共有2层。所以总数是3×2×2=12个立方体。

• Count the cubes in one layer first.
• 先数一层的立方体数量。
• Then multiply by the number of layers.
• 然后乘以层数。
• Each cube represents 1 cubic unit of volume.
• 每个立方体代表1立方单位的体积。

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4. Volume of a Rectangular Prism | 长方体的体积

A rectangular prism has length (l), width (w), and height (h). To find its volume, multiply the three dimensions together.

长方体有长（l）、宽（w）和高（h）。要求它的体积，只需把这三个维度相乘。

Volume = length × width × height

体积 = 长 × 宽 × 高

For example, a box is 5 cm long, 4 cm wide, and 3 cm high. Its volume is 5 × 4 × 3 = 60 cm³. Notice that the order of multiplication does not matter; 4 × 5 × 3 also gives 60.

例如，一个盒子长5厘米，宽4厘米，高3厘米。它的体积是5×4×3=60 cm³。注意乘法的顺序不重要；4×5×3同样得到60。

This formula works because length × width gives the area of the base (the bottom face), and then multiplying by the height counts how many layers of unit cubes fit inside.

这个公式之所以成立，是因为长×宽得到底面的面积，再乘以高，就数出能放多少个单位立方体的层数。

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5. Volume of a Cube | 立方体的体积

A cube is a special rectangular prism where all sides are equal. If one side length is s, then the volume is s × s × s, which we write as s³.

立方体是一种特殊的长方体，所有棱长都相等。如果一条棱长为s，那么体积就是s×s×s，记作s³。

Volume of a cube = s × s × s = s³

立方体的体积 = s × s × s = s³

If a cube has side length 6 cm, then its volume is 6 × 6 × 6 = 216 cm³. Make sure to write the correct cubic unit after the number.

如果一个立方体的棱长是6厘米，那么它的体积是6×6×6=216 cm³。记得在数字后面写上正确的立方单位。

• Use the same unit for length, width, and height before multiplying.
• 在相乘之前，确保长、宽、高使用相同的单位。
• The exponent ³ means the number is used as a factor three times.
• 指数³表示这个数相乘三次。

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6. Finding a Missing Dimension | 求未知维度

If we know the volume of a rectangular prism and two of its dimensions, we can find the third dimension by dividing.

如果我们已知长方体的体积以及其中两个维度，就可以通过除法求出第三个维度。

When a prism has volume 120 cm³, length 10 cm, and width 4 cm, we can find the height using:

当一个长方体的体积是120 cm³，长10厘米，宽4厘米时，我们可以这样求高：

Height = Volume ÷ (length × width)

高 = 体积 ÷ (长 × 宽)

First, calculate 10 × 4 = 40. Then 120 ÷ 40 = 3. So the height is 3 cm. We can check: 10 × 4 × 3 = 120 cm³.

先计算10×4=40，然后120÷40=3。所以高是3厘米。我们可以验算：10×4×3=120 cm³。

This idea helps us solve real problems, such as finding how deep a box must be to hold a certain volume.

这个思路帮助我们解决实际问题，比如求一个盒子要多深才能容纳一定的体积。

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7. Volume of Composite Figures | 组合图形的体积

Some solid figures are made of two or more rectangular prisms. These are called composite figures. To find the total volume, split the figure into simpler prisms, find each volume, and then add them together.

有些立体图形由两个或更多长方体组成，这些称为组合图形。要求总体积，先把图形分割成更简单的长方体，分别求体积，然后把它们相加。

Imagine an L-shaped block made from two rectangular prisms: one prism is 4 cm × 3 cm × 2 cm and the other is 5 cm × 2 cm × 2 cm. The first volume is 24 cm³, and the second is 20 cm³. The total volume is 24 + 20 = 44 cm³.

想象一个L形积木由两个长方体组成：一个长方体的尺寸是4 cm×3 cm×2 cm，另一个是5 cm×2 cm×2 cm。第一个体积是24 cm³，第二个是20 cm³。总体积是24+20=44 cm³。

• Look for lines where the shape can be split into prisms.
• 寻找可以把图形分割成长方体的分界线。
• Calculate each part separately.
• 分别计算每一部分的体积。
• Add the volumes together.
• 把各部分体积相加。

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8. Solving Word Problems with Volume | 用体积解决应用题

Word problems often describe a container, a room, or a block of material. Read carefully to identify the length, width, and height, and decide whether you need to multiply or divide.

应用题经常描述容器、房间或一块材料。仔细阅读以确定长、宽、高，并判断需要使用乘法还是除法。

Example: A fish tank is 40 cm long, 25 cm wide, and 30 cm high. What is the volume?

例：一个鱼缸长40厘米，宽25厘米，高30厘米。它的体积是多少？

Volume = 40 × 25 × 30 = 30,000 cm³. This is the amount of space inside the tank.

体积=40×25×30=30,000 cm³。这就是鱼缸内部的空间大小。

In some problems, we must convert units before multiplying. If a cube has side length 2 m, its volume is 8 m³, which equals 8,000,000 cm³ because 1 m = 100 cm and 100 × 100 × 100 = 1,000,000.

有些问题需要先换算单位再相乘。如果一个立方体的棱长是2米，它的体积是8 m³，等于8,000,000 cm³，因为1米=100厘米，且100×100×100=1,000,000。

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9. Volume and Capacity | 体积与容量

Capacity is the amount of liquid a container can hold. It is closely related to volume. A cube with volume 1 cm³ can hold 1 millilitre (mL) of water. So 1 cm³ = 1 mL.

容量是容器能容纳液体的量，它与体积密切相关。一个体积为1 cm³的立方体可以装1毫升（mL）水。所以1 cm³ = 1 mL。

For larger containers, we use litres. Since 1 L = 1,000 mL, a box with volume 2,000 cm³ has a capacity of 2 L. This connection helps us in science and cooking.

对于更大的容器，我们使用升。因为1升=1,000毫升，所以体积为2,000 cm³的盒子容量为2升。这个联系在科学和烹饪中很有用。

Volume | 体积	Capacity | 容量
1 cm³	1 mL
1,000 cm³	1 L
1 m³	1,000 L

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10. Summary and Practice Tips | 总结与练习建议

To find the volume of a rectangular prism, use the formula length × width × height. For a cube, use side × side × side. For composite figures, split, calculate, and add. Always write the answer in cubic units.

求长方体的体积使用公式长×宽×高；求立方体的体积使用棱长×棱长×棱长；对于组合图形，先分割、再计算、最后相加。答案一定要写成立方单位。

When solving, check these three things: Are all units the same? Did I multiply the correct numbers? Is my answer reasonable? Checking your answer by reversing the operation can prevent mistakes.

解题时请检查三件事：所有单位是否一致？我乘的数值是否正确？答案是否合理？通过逆运算检查答案可以避免错误。

Practice with real objects: measure a book, a shoe box, or a lunch box, and compute its volume. The more you practise, the easier it becomes to visualise space and make accurate calculations.

用真实物体练习：测量一本书、一个鞋盒或餐盒，并计算其体积。练习越多，就越容易想象空间并做出准确的计算。

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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