如果正方形的周长和面积相等，请计算正方形的边。

正方形是一个有四个边和四个角的封闭二维图形。所有四个边的长度相等且彼此平行。

正方形是四边形，其中：

• 对边是平行的。
• 四边相等。
• 所有角度均为 90°。

正方形的周长

正方形边界的长度可以通过对其所有边进行求和来计算，由下式给出，

正方形的周长=边+边+边+边。

因此，正方形的周长 = (4 × Side) 单位

正方形面积

正方形的面积是包围在其边界内的空间。它由其两侧中任何一条的乘积给出，并以平方单位表示。通过公式，我们有，

正方形的面积 = 边 × 边

让我们假设正方形的边等于 S。

所以，

面积 = S ² 。

如果正方形的周长和面积相等，请计算正方形的边。

解决方案：

Since, we know,

⇒ Perimeter of a square

Perimeter of a square = Side + Side + Side + Side

Perimeter of a square = 4 × Side  ……….(1)

⇒ Area of a square

Area of a square = Side × Side

Area of a square = (Side)²  ……….(2)

Let us assume the side of the square to be ‘s’.

Further, if we equate (1) and (2) as per the problem

We will get;

Perimeter of a square = Area of a square …..(given)

⇒ 4 × Side = Side × Side

⇒ 4 × s = s × s

⇒ 4 × s = s²

Taking L.H.S ‘s’ to R.H.S

⇒ 4 = $\frac{s^2}{s}$

⇒ 4 = s

⇒ s = 4

Side = 4 unit

Therefore,

If the side of the square is 4 units, then its perimeter and area will be equal

On checking square with 4 units side will give equal perimeter and area.

⇒ Perimeter of a square = 4 × Side  

Perimeter of a square = 4 × 4

Perimeter of a square = 16 unit

⇒ Area of square = Side × Side

Area of square = 4 × 4

Area of square = 16 unit²

Hence,

We can see that if the side of the square is 4 units, then the perimeter and area of the square are equal.

示例问题

问题 1. 求边长为14 m的正方形的周长和面积？

解决方案：

Here we have to find the perimeter and area of a square

As we know that

⇒ Perimeter of a square = 4 × Side  

Perimeter of a square = 4 × 14

Perimeter of a square = 56 m

⇒ Area of square = Side × Side

Area of square = 14 × 14

Area of square = 196 m²

Therefore,

The perimeter of a square with a side of 14 m is 56 m 

The area of a square with a side of 14 m is 196 m²

问题 2. 假设一个正方形地面的周长是 80 m。然后求正方形地面的面积？

解决方案：

Here we have to find the area of the square ground

Given:

Perimeter of the square ground = 80 m

As we know that

Perimeter of a square = 4 × side

Perimeter of a square = 4 × s

80 = 4 × s

s = $\frac{80}{4}$

s = 20 m

Side of the square ground is 20 m.

Now,

Area of a square = Side × Side

Area of a square = s × s

Area of a square = 20 × 20

Area of a square = 400 m²

Therefore,

The area of the square with a perimeter of 80 m is 400 m²

问题 3. 如果正方形的面积是 121 cm ² ，求正方形的周长？

解决方案：

Here we have to find the perimeter of the square

Given:

Area of the square = 121 cm²

As we know that

Area of the square = Side × Side

Area of the square = s × s

Area of the square = s²

121 = s²

s = $\sqrt{121}$

s = 11 cm

Side of the square = 11 cm

Further,

Perimeter of the square = 4 × side

Perimeter of the square = 4 × 11

Perimeter of the square = 44 cm

Therefore,

Perimeter of the square with an area of 121 cm² is 44 cm.

问题 4. 如果正方形地板的周长是 88 m，那么以每米 20 卢比的比率计算铺地板的成本？

解决方案：

Here we need to find the cost of carpeting the floor

Given:

Perimeter of the floor = 88 m

As we know that

Perimeter of the square = 4 × side

Perimeter of the square floor = 4 × s

88 m = 4 × s

s = $\frac{88}{4}$

s = 22 m

The side of the square is 22 m

Further,

Carpet required = Area of the floor

Area of the square floor = s × s

Area of the square floor = 22 × 22

Area of the square floor = 484 m²

Now,

Cost of carpeting = Area of the square floor × ₹20

Cost of carpeting = 484 × 20

Cost of carpeting = ₹9680

Therefore,

The cost of carpeting the square floor is ₹9680.

问题 5. 计算边长为44 厘米的方形盒子的周长和面积？

解决方案：

Here we have to find the perimeter and area of a square box

As we know that

⇒ Perimeter of a square = 4 × Side  

Perimeter of a square box = 4 × 44

Perimeter of a square box = 176 cm

⇒ Area of square = Side × Side

Area of square box = 44 × 44

Area of square = 1936 cm²

Therefore,

The perimeter of a square with a side of 44 m is 176 cm

The area of a square with a side of 44 m is 1993 cm².