当周长增加一倍时，正方形的面积会发生什么变化？

正方形是由四个相等的边组成的几何图形。它是一个规则的四边形，其中所有的角度都是直角。基本上，它可以被认为是矩形的一种特殊情况，其中所有边的长度都相等。例如，一个方形的庭院。

正方形的性质

• 正方形的四个边相等。
• 正方形的对角线相等。
• 正方形的四个角都相等。
• 正方形的对角线平分它的角。

正方形的周长

正方形的周长称为包围几何图形的边界的总长度。

Perimeter of square = Sum of all sides of a square

让我们假设 s 是正方形的边。

因为，我们知道，正方形的所有边都是相等的。

Perimeter of square = s + s + s + s

Perimeter of square = 4s ….. (I)

正方形的面积

正方形的面积是包围在其边界内的空间。每个都以平方米为单位进行描述。

Area of square = Product of any two sides

让我们假设 s 是正方形的边。

因为，我们知道，正方形的所有边都是相等的。

Area of square = s * s

Area of square = s² ….. (I)

当周长增加一倍时，正方形的面积会发生什么变化？

解决方案：

Now, 

Let us assume P to be the perimeter of the square. Now, as proved above,

P = 4s, where s is the side of the square….I

According to the question, 

Perimeter is doubled. Let us assume the new perimeter to be P’ and s’ to be the new side of the square. 

P’ = 2P

Substituting values of P from I, we have

P’ = 2 (4s)

P’ = 8s

Therefore, 

s = P’/8

Now, 

P’ = 4s’

Substituting the values, 

s = 4s’/8

s = s’/2

s’ = 2s 

Therefore, the side becomes twice the original side. 

Now,

Old Area, A = s²

New Area, A’ = s’²

Substituting values of s’ from II,we get, 

A’ = (2s)²

A’ = 4s²

Therefore, 

A’ = 4A

Thus, area becomes 4 times of the original area. 

示例问题

问题 1. 如果正方形的周长是 16 厘米，那么求正方形的面积，如果周长增加一倍，正方形的面积会增加多少倍？

解决方案：

Here we have the perimeter of square 16 cm

As we know that

Perimeter of square = 4 × side (s)

16 = 4 × side

s = 16/4

s = 4 cm

Side of a square is 4 cm

Further,

Finding the area of square 

Area of square = Side × Side

Area of square = 4 × 4

Area of square = 16 cm²

Now,

When the perimeter of the square is doubled

2 × Perimeter of the square = 2 × 16

= 32 cm

Now,

Side of the doubled perimeter = s’

32 = 4 × s’

s’ = 32/4

s’ = 8 cm

Further new area

Area of doubled perimeter square = s’ × s’

Area of double perimeter square = 8 × 8

Area of double perimeter square = 64 cm²

To find how many times the area of square increased after doubling the perimeter

= $\frac{Area\ of\ square\ with\ doubled\ perimeter}{Area\ of\ square}$

= 64/16

= 4 times

Therefore,

Area of the square is increased by 4 times after doubling the perimeter of the square.

问题 2. 如果正方形的周长增加一倍，求正方形的面积增加多少倍？如果正方形的边长为 10 m。

解决方案：

Here we have the side of square 10 m

As we know that

Perimeter of a square = 4 × side (s)

Perimeter of a square  = 4 × 10

Perimeter of a square = 40 m

Perimeter of a square = 40 m

Perimeter of a square = 40 m

Further,

Finding the area of square

Area of square = Side × Side

Area of square = 10 × 10

Area of square = 100 m²

Now,

When the perimeter of the square is doubled

2 × Perimeter of the square = 2 × 40

= 80 m

Now,

Side of the doubled perimeter = s’

80 = 4 × s’

s’ = 80/4

s’ = 20 m

Further new area

Area of doubled perimeter square = s’ × s’

Area of double perimeter square = 20 × 20

Area of double perimeter square = 400 m²

To find how many times the area of square increased after doubling the perimeter

= $\frac{Area\ of\ square\ with\ doubled\ perimeter}{Area\ of\ square}$

= 400/100

= 4 times

Therefore,

Area of the square is increased by 4 times after doubling the perimeter of the square.

问题 3. 如果正方形的周长增加三倍，请计算正方形的面积增加了多少倍？正方形的周长是 20 厘米。

解决方案：

Here we have the perimeter of square 20 cm

As we know that

Perimeter of square = 4 × side (s)

20 = 4 × side

s = 20/4

s = 5 cm

Side of a square is 5 cm

Further,

Finding the area of square

Area of square = Side × Side

Area of square = 5 × 5

Area of square = 25 cm²

Now,

When the perimeter of the square is tripled

3 × Perimeter of the square = 3 × 20

= 60 cm

Now,

Side of the doubled perimeter = s’

60 = 4 × s’

s’ = 60/4

s’ = 15 cm

Further new area

Area of tripled perimeter square = s’ × s’

Area of tripled perimeter square = 15 × 15

Area of tripled perimeter square = 225 cm²

To find how many time area of square increased after tripling the perimeter

= $\frac{Area\ of\ square\ with\ tripled\ perimeter}{Area\ of\ square}$

= 225/25

= 9 times

Therefore,

Area of the square is increased by 9 times when the perimeter of the square is tripled.

问题 4. 正方形的周长是 40 m。如果正方形的周长增加了 4 倍，那么求正方形的面积增加了多少倍？

解决方案：

Here we have the perimeter of square 40 cm

As we know that

Perimeter of square = 4 × side (s)

40 = 4 × side

s = 40/4

s = 10 m

Side of a square is 10 m

Further,

Finding the area of square

Area of square = Side × Side

Area of square = 10 × 10

Area of square = 100 m²

Now,

When the perimeter of the square is 4 times,

4 × Perimeter of the square = 4 × 40

= 160 m

Now,

Side of the doubled perimeter = s’

160 = 4 × s’

s’ = 160/4

s’ = 40 m

Further new area

Area of 4 times perimeter square = s’ × s’

Area of 4 times perimeter square = 40 × 40

Area of 4 times perimeter square = 1600 m²

To find how many times the area of the square increased after 4 times the perimeter

= $\frac{Area\ of\ square\ with\ 4\ times\ perimeter}{Area\ of\ square}$

= 1600/100

= 16 times

Therefore,

Area of the square is increased by 16 times when the perimeter of the square is 4 times.

问题 5.如果正方形的面积是 400 cm ² ，求它的周长？

解决方案：

Area of square = 400 cm²

side × side = 400

s² = 400

s = √400

s = 20 cm

Side of the square = 20 cm

Perimeter of square = 4 × side

Perimeter of square = 4 × 20

Perimeter of square = 80 cm