如何求正方形和等边三角形的周长？

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)

示例：求每边为 10 个单位的正方形的周长。

解决方案：

Here we have been given each side of square is 10 units

As we know that,

Perimeter of square, (P) = Side + Side + Side + Side

Perimeter of square, (P) = 10 + 10 + 10 + 10

Or

Perimeter of square, (P) = 4 × Side

Perimeter of square, (P) = 4 × 10

Therefore,

Perimeter of square, (P) = 40 units

等边三角形

等边三角形，也称为等角三角形，是三角形的一种特殊情况，其中所有三个边的长度都相等。这些边在拐角处也对向相等的角度。边长的总和得出三角形的总周长。

等边三角形的性质

等边三角形具有以下特性：

• 三边相等。
• 所有三个角度都等于 60°

在给定的图中，三角形 ABC 的边相等，即 AB = BC = CA。

等边三角形的周长公式

让我们假设 a 是等边三角形的边。

现在，

Perimeter of equilateral triangle = Sum of all sides 

= a + a + a 

Now, the perimeter of an equilateral triangle = 3a 

示例问题

问题 1. 计算正方形的周长，对角线为 20 个单位。

解决方案：

Here we have to calculate the perimeter of the square using the given diagonal dimension,

To find the perimeter of the square using diagonal length use the formula below,

Perimeter of square = 4 × √2 ($\frac{Diagonal}{2}$)

Here diagonal = 20 units

By applying value of diagonal in the above formula

Perimeter of square =  $4\times\sqrt2\times\frac{20}{2}$

Perimeter of square = 40√2

问题 2. 如果正方形的面积为 64 m ² ，请计算其周长？

解决方案：

Here we have to calculate the perimeter of the square using the given area,

To find the perimeter of the square using the area of square use the formula below,

Perimeter of square =  $4\times\sqrt{Area}\ units$

Here Area = 64 m²

By applying the value of the area in the above formula

Perimeter =  4 × √64

Perimeter of square = 4 × 8

Perimeter of square = 32 m

问题 3. 假设一个等边三角形的面积是 64√3 cm ² ，求它的周长？

解决方案：

Here we have to find the perimeter of equilateral triangle using its area

Formula for equilateral triangle area = $\frac{\sqrt3}{4}a^2$

Area of equilateral triangle = 64√3

64√3 = $\frac{\sqrt3}{4}a^2$

a² = $64\sqrt3\times\frac{4}{\sqrt3}$

a = √256

a = 16 

Therefore,

Side of the equilateral triangle is 16 cm

Now further finding perimeter of equilateral triangle

Perimeter of equilateral triangle = side + side + side = 3a

Perimeter of equilateral triangle = 3 × 16

Perimeter of equilateral triangle = 48 cm

问题 4. 计算一个等边三角形的周长，三角形的高是 16√3 厘米。

解决方案：

Here we have to find the Perimeter of the equilateral triangle with the height 16√3 cm

Formula for calculating perimeter using height is given below

Height = $\frac{\sqrt3}{2}a$

Here a is the side of the equilateral triangle

16√3 = $\frac{\sqrt3}{2}a$

a = $16\sqrt3\times\frac{2}{\sqrt3}$

a = 32 cm

Now further finding perimeter of the equilateral triangle

Perimeter of equilateral triangle = side + side + side = 3a

Perimeter of equilateral triangle = 3 × 32

Perimeter of equilateral triangle = 96 cm

问题 5. 如果等边三角形的周长和正方形的周长相同，等边三角形的边是 15 厘米，那么求正方形的周长和边长？

解决方案：

Here we have to find the perimeter of the square using the given side of an equilateral triangle and we are given that the perimeter of the square and triangle are the same

First, we will find the perimeter of the equilateral triangle

Perimeter of equilateral triangle = side + side + side = 3a

The perimeter of equilateral triangle = 3 × 15

The perimeter of equilateral triangle = 45 cm

According to the question

The perimeter of equilateral triangle = Perimeter of a square

Therefore,

Perimeter of square = 45 cm

Perimeter of square = side + side + side + side = 4 × side = 45 cm

4 × side = 45 cm

Side = $\frac{45}{4}$

Side = 11.25 cm