如果立方体的边缘减少到四分之一，它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少到四分之一，它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 one fourth

This implies that,

a’ = a/4

Volume of the cube, V’ = $\left(\frac{a}{4}\right)^3$

On simplifying, we get,

$=\frac{a^3}{64}\\ =\frac{1}{64}a^3\\ =\frac{1}{64}V$

The volume of the Cube when the edge is reduced to one fourth is $\frac{1}{64}$ times the original volume.

示例问题

问题 1. 立方体的边减少多少倍，立方体的体积减少了多少倍$\frac{1}{8}$ ?

解决方案：

Here we have to find how much volume will be reduced if the edge of a cube is reduced by $\frac{1}{8}$

Assume a is the side of the cube

If the side of cube is reduced by $\frac{1}{8}$ then,

a =  $\frac{1}{8}\times a$ = $\frac{a}{8}$

As we know that,

Volume of cube V =  a × a × a = a³

Volume of the reduced edge cube V’ = $\frac{a}{8}\times\frac{a}{8}\times\frac{a}{8}$

Volume of the reduced edge cube V’ = $\left(\frac{a}{8}\right)^3$

Volume of the reduced edge cube V’ = $\frac{a^3}{512}$

Volume of the reduced edge cube V’ = $\frac{1}{512}\times a^3$

Volume of the reduced edge cube V’ = $\frac{1}{512}\times V$

Therefore,

We can see that if we reduce the edge of the cube by $\frac{1}{8}$ the volume of the cube is reduced by $\frac{1}{512}$ times.

问题 2. 如果立方体的边增加四倍，它的体积会增加多少？

解决方案：

Here we have to find the change in the volume of the cube when its edge is increased by four times.

Assume a is the side of the cube

If the side of the cube is four times

a = 4 × a = 4a

As we know that,

Volume of cube V =  a × a × a = a³

Volume of cube with four times side V’ =  4a × 4a × 4a

Volume of cube with four times side V’ =  64a³

Volume of cube with four times side V’ =  64 × V

Therefore,

We can see that if we increase the edge of the cube by four times its volume is increased by 64 times.

问题 3. 如果我们将立方体的边缘减半，立方体的体积会减少多少？

解决方案：

Here we have to find how much volume will be reduced if the edge of a cube is reduced by halve

Assume a is the side of the cube

If the side of the cube is reduced by halve then,

a = $\frac{1}{2}\times a$

= $\frac{a}{2}$

As we know that,

Volume of cube V =  a × a × a = a³

Volume of the reduced edge cube V’ = $\frac{a}{2}\times\frac{a}{2}\times\frac{a}{2}$

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

Volume of the reduced edge cube V’ = $\frac{a^3}{8}$

Volume of the reduced edge cube V’ = $\frac{1}{8}\times a^3$

Volume of the reduced edge cube V’ = $\frac{1}{8}\times V$

Therefore,

We can see that if we reduce the edge of the cube by half the volume of the cube is reduced by $\frac{1}{8}$ times.

问题 4. 如果立方体的体积是 1728 cm ³ 。然后找到立方体的侧面。

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 1728 cm³

1728 = a × a × a = a³

1728 = a³

a = $\sqrt[3]{1728}$

a = 12 cm

Therefore,

Side of the cube with volume 1728 cm² is 12 cm.

问题 5. 一个立方体的盒子必须装满边长 15 厘米的沙子。找出填充立方体所需的沙子的体积？

解决方案：

Here we have to find the sand that can be filled in the cubical box

We are given that the side of the room is 15 cm

As we know that

Volume of the cube = a × a × a

Volume of the cube = a³

Volume of the cube = 15³

Volume of the cube = 3375 m³

Therefore,

3375 m³ sand can be filled in a cubical box of size 15 cm.