如果长方体的所有尺寸都加倍，长方体的体积和表面积会发生什么变化？

长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。一个长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。在长方体的情况下，长度、宽度或高度可能相等，也可能不相等。

长方体的性质

• 所有的面本质上都是矩形的。
• 所有的角度都是直角
• 长方体的相对面相等。

长方体的体积

长方体的体积等于图形中占据的空间量。任何三维图形的体积取决于三个边的长度，即它的长、宽和高。它可以被认为是一个实心矩形。让我们假设长方体的高度为 h，l 其长度和宽度分别用 b 个单位表示。

除此之外，让我们假设 V 是长方体的体积。推导出其公式，

长方体的体积=底面积×高

长方体的底面积 = l × b

因此，

Volume of a cuboid, V = Product of all three sides = l × b × h = lbh

长方体的表面积

长方体由以下两种表面积表示：

• 总表面积——每个面的面积之和。用 S 表示。

Total Surface Area of Cuboid, S = 2 × (lb + bh + lh)

• 侧表面积——每个面的面积之和，不包括底部和顶部。用 L 表示。

总表面积和侧表面积可以用长方体的长度 (l)、宽度 (b) 和高度 (h) 表示为：

Lateral Surface Area of Cuboid, L = 2h (l + b)

如果长方体的所有尺寸都加倍，长方体的体积和表面积会发生什么变化？

解决方案：

让我们假设 l、b、h 分别是第一个长方体的长、宽、高。

让我们假设 L、B、H 是新形成的长方体的长、宽、高。

由于长方体的所有边都是双倍的，因此，我们有，

我们知道，

L = 2l

B = 2b

H = 2h

表面积的计算

Surface area of the first cuboid S = 2(lb + bh + hl)

Now, 

Surface area of new cuboid S, 

= 2(LB + BH + HI)

= 2[(2l)(2b) + (2b)(2h) + (2h)(2l)]

= 2(4lb + 4bh + 4hl)

= 4[2(lb + bh + hl)]

= 4S

∴  The surface area of the new cuboid is 4S.

体积计算

Volume of the original cuboid is given by V = l × b × h

When its length is doubled, its length becomes 2 × l.  

When its height is double, it becomes 2 × h.  

When its breadth is double, it becomes 2 × b.  

Now, Volume of the new cuboid = length × breadth × height

= 2 × l × 2 × b × 2 × h

= 8 × l × b × h

∴ Therefore, the volume of the new cuboid is eight times the initial volume.

示例问题

问题1.如果长方体的长度加倍，高度加倍，宽度相同，计算长方体的体积？

解决方案：

Assume that,

Length of cuboid = l

Height of cuboid = h

Breadth of cuboid = b

Therefore,

Volume of cuboid = Length × Breadth × Height

= l × b × h

According to the question

Length of cuboid is doubled = 2 × l = 2l

Height of cuboid is doubled = 2 × h = 2h

Breadth remains same

Further,

Volume of the cuboid with the new dimensions = Length × Breadth × Height

= 2 × l × b × 2 × h

= 4 × l × b × h

Hence,

We can see that the volume of the cuboid with the new cuboid is four times the volume of the initial cuboid.

问题 2. 长方体的长翻倍，高相同，宽减半，求长方体的体积？

解决方案：

Assume

Length of a cuboid = l 

Breadth of a cuboid = b 

Height of a cuboid = h 

Therefore, 

Volume of a cuboid = Length × Breadth × Height 

= l × b × h

Further,

Length of cuboid is doubled = 2 × l = 2l 

Breadth of cuboid is halved  = $\frac{b}{2}$ 

Height of cuboid is same = h 

Therefore,

Volume of new dimension cuboid = Length × Breadth × Height 

= 2 × l × $\frac{b}{2}$ × h 

= l × b × h 

Hence,

We can clearly see that after Length is doubled, height is the same and breadth is halved the volume of the cuboid remains the same.

问题 3. 假设一个长方体鞋盒的表面积是 126 cm ² 。给定长方体长6厘米，长方体高3厘米，求长方体鞋盒的宽度。

解决方案：

Here we are given that,

Surface area of the cuboid shoe box = 126 cm²

Length of the cuboid shoe box = 6 cm

Height of the cuboid shoe box = 3 cm

Assume Breadth of the cuboid shoe box = b cm

Thus,

⇒ 126 cm² = 2 × {(Length × Breadth) + (Breadth × Height) + (Length × Height)}

⇒ 126 cm² = 2 × {(6 × b) + (b × 3) + (6 × 3)}

⇒ 126 cm² = 2 × (6b + 3b + 6 x 3)

⇒ 126 cm² = 2 × (9b + 18)

⇒ 126 cm² = 18b + 36

⇒ 18b = 126 – 36

⇒ 18b = 90

⇒ b = $\frac{90}{18}$

⇒ b = 5 cm

Therefore.

Breadth of the cuboid show box is 5 cm.

问题 4. 如果长方体的长、宽、高三倍，长方体的表面积会发生什么变化？

解决方案：

Assume

Length of cuboid = l

Breadth of cuboid = b

Height of cuboid = h

Surface Area of Cuboid = 2 × {(Length × Breadth) + (Breadth × Height) + (Length × Height)}

According to the question

Length of cuboid tripled = 3 × length = 3l

Breadth of cuboid tripled = 3 × breadth = 3b

Height of cuboid tripled = 3 × height = 3h

Surface Area of a increased dimensional Cuboid = 2 × (3l × 3b + 3b × 3h +3l × 3h)

= 9 × 2 × (lb + bh + lh)

= 9 × Surface area of initial cuboid

Therefore,

We can see that if we triple all the dimensions of the cuboid then its surface area becomes 9 times. 

问题 5. 如果一个长方体的体积是 24 cm ³ 。如果所有维度都加倍，那么找到长方体的体积？

解决方案：

Here we are given that,

Volume of cuboid = 24 cm³

According to the question

If we double all the dimensions of the cuboid

Then,

Volume of the cuboid =  24 × 2 × 2 × 2

= 192 cm²