如果立方体的边缘减半，立方体的体积会发生什么变化？

立方体是一个三维立体物体。它也被称为正六面体，是五个柏拉图立体之一。所有的边至少共享一条公共边。立方体的结构可以这样定义：每个面都连接到四个顶点和四个边，顶点连接到三个边和三个面，并且边连接到两个面和两个顶点。

立方体的属性

• 所有的正方形面都是相等的。
• 所有边的长度相等。
• 由八个顶点和十二条边组成。
• 长度 = 宽度 = 高度

立方体体积

立方体占据的总体积称为它的体积。由于立方体具有所有相似的面，因此，边的长度也是相等的。因此，立方体的长、宽、高是等长的。立方体的体积是根据几何图形完全占据的立方单位来定义的。立方体的体积可以用厘米³ ，米³的形式表示

立方体的体积可以使用立方体边缘的长度来计算。让我们假设立方体的边是a。我们还假设 V 是立方体的体积。所以，立方体公式的体积是：

Volume of a Cube = Length × Width × Height

Since, all the sides are equal in cube.

Volume = a × a × a

Volume = a³

如果立方体的边缘减半，立方体的体积会发生什么变化？

回答：

The formula used in solving this problem is, 

Volume of the Cube,  V = a³

Let us assume the side of the cube to be a.

Let us assume the volume of the cube to be ‘V’. 

Now, we know, 

The edge of the cube is reduced to half. 

This implies that,

a’ = a/2

Volume of the cube, V’ = $\left(\frac{a}{2}\right)^3$ .

On simplifying, we get, 

$=\frac{a^3}{8}\\ =\frac{1}{8}a^3\\ =\frac{1}{8}V$

The volume of the Cube when the edge is halved is $\frac{1}{8}$ times the original volume.

示例问题

问题1.如果立方体的边缘翻倍，立方体的体积会发生什么变化？

解决方案：

Here we have to find the change in the volume of the cube when its edge is doubled

Assume a is the side of the cube

If the side of the cube is doubled

a = 2 × a = 2a

As we know that,

Volume of cube V =  a × a × a = a³

Volume of the cube of doubled side V’ =  2a × 2a × 2a

Volume of the cube of doubled side V’ =  8a³

Volume of the cube of doubled side V’ =  8 × V

Therefore,

We can see that if we double the edge of the cube its volume becomes 8 times.

问题2.如果立方体的体积是8000 m ³ ，那么求立方体的边

解决方案：

Here we have to find the length of the side of the cube,

Assume the side of the cube is ‘a’

As we know that,

Volume of cube  =  a × a × a = a³

In the question, we are given the volume of the cube as 8000 m³

8000 = a × a × a = a³

8000 = a³

a = $\sqrt[3]{8000}$

a = 20 m

Therefore,

Side of the cube with a volume of 8000 m³ is 20 m.

问题 3. 求一个边长为 30 厘米的立方体的体积增加，如果它的边增加 3 倍？

解决方案：

Here we have to find the percentage increase in the volume of 2 cubes 

Assume a is the side of the cube

As we know that

Volume of cube V = a × a × a = a³

V = 30 × 30 × 30 = 27000 cm³

Now assume the side of increased side cube be a’

a’ = 3 × a = 3a

As we know that

Volume of cube V’ = 3a × 3a × 3a

V’ = 27a³

V’ = 27 × V

Therefore,

We can see that if we increase the side of the cube by 3 times then its volume will be increased by 27 times.

问题 4. 如果将两个边长为 15 厘米的立方体连接起来形成一个长方体，那么求两个立方体连接起来形成的长方体的体积？

解决方案：

We can calculate the volume of the cuboid formed by joining two cubes by two methods

Method 1. By the formula of volume of a cuboid, in which the sides of cuboid will be; 

Length = 15 + 15 = 30 cm

breadth = 15 cm

height = 15 cm

Method 2. By finding the volume of single cube and then multiplying it by 2.

Find by using the formula of volume of cuboid,

Volume of Cuboid = Length × Breadth × Height

= l × b × h

= 30 × 15 × 15

= 6750 cm³

Find the volume by using volume of cube formula and then multiplying by 2.

Volume of the cube = a × a × a 

= a³

= 15³

= 3375 cm³

Now multiply by 2,

= 2 × 3375

= 6750 cm³

问题 5. 找出边长为 10 m 的立方体房间可以填充多少空气？

解决方案：

Here we have to find the air that can be filled in the cubical room

We are given that the side of the room is 10 m

As we know that

Volume of the cube = a × a × a 

Volume of the cube = a³

Volume of the cube = 10³

Volume of the cube = 1000 m³

Therefore,

1000 m³ air can be filled in a cubical room of size 10 m.