如何在给定周长的情况下找到等边三角形的高度？

等边三角形是所有三个边相等的几何二维图形。所有边缘在拐角处对向 60° 角。因此，三角形所有角的和为180°。

等边三角形的性质

• 等边三角形的三个边相等。
• 所有三个角度都相等，即各 60°。
• 正多边形有三个相等的边。

等边三角形的周长

正方形的周长称为包围几何图形的边界的总长度。

等边三角形的周长=等边三角形所有边的总和

让我们假设 a 是等边三角形的边。

因为，我们知道，等边三角形的所有边都是相等的。

等边三角形的周长 = a + a + a

等边三角形的周长 = 3a

等边三角形的高度

等边三角形的周长可以用三角形的高来计算。

让我们假设“a”是等边三角形的边。

自从，

等边三角形的高度 = $\frac{\sqrt3}{2}a$

现在，让我们假设 P 是等边三角形的周长。

P = 边 + 边 + 边

因为，所有的边都是相等的，

P = 3a

替换值，

高度 = $\frac{\sqrt3}{2}(\frac{P}{3})$

示例问题

问题 1. 计算一个高度为 20√3 厘米的等边三角形的周长。

解决方案：

Here we have to find the Perimeter of the equilateral triangle with the height 20√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 

20√3 = $\frac{\sqrt3}{2}a$

a = $20\sqrt3\times\frac{2}{\sqrt3}$

a = 40 cm

Now further finding perimeter of the equilateral triangle

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

Perimeter of equilateral triangle = 3 × 40

Perimeter of equilateral triangle = 120 cm

问题 2. 假设一个等边三角形的高是 100√3 m，那么求它的周长？

解决方案：

Here we have to find the Perimeter of the equilateral triangle with the height 100√3 m

Formula for calculating perimeter using height is given below

Height = $\frac{\sqrt3}{2}a$

Here a is the side of the equilateral triangle

100√3 = $\frac{\sqrt3}{2}a$

a = $100\sqrt3\times\frac{2}{\sqrt3}$

a = 200 m

Now further finding perimeter of the equilateral triangle

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

Perimeter of equilateral triangle = 3 × 200

Perimeter of equilateral triangle = 600 m

问题 3. 求一个边长为 22 m 的等边三角形的周长。另外，找到它的高度？

解决方案：

First find the perimeter of an equilateral triangle

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

Perimeter of equilateral triangle = 3 × 22

Perimeter of equilateral triangle = 66 m

Further finding the height of the equilateral triangle

Height = $\frac{\sqrt3}{2}a$

Here a is side of the equilateral triangle

Height = $\frac{\sqrt3}{2}\times22$

Height of the equilateral triangle = 11√3

问题 4. 如果等边三角形的周长是 99 厘米，那么求等边三角形的高？

解决方案：

First we have to find the side of equilateral triangle

As we know that

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

As given in the question

Perimeter of equilateral triangle = 99 cm

3a = 99 cm

a = $\frac{99}{3}$

a = 33 cm

Hence the side of triangle is 33 cm

Further finding the height of the equilateral triangle

Height = $\frac{\sqrt3}{2}a$

Here a is side of the equilateral triangle

Height = $\frac{\sqrt3}{2}\times33$

Height of the equilateral triangle = 16.5√3

问题 5. 找出一个高200 厘米的等边三角形的周长。

解决方案：

Here we have to find the Perimeter of the equilateral triangle with the height 200 cm

Formula for calculating perimeter using height is given below

Height = $\frac{\sqrt3}{2}a$

Here a is the side of the equilateral triangle

200 = $\frac{\sqrt3}{2}a$

a = $200\times\frac{2}{\sqrt3}$

a = 230.94 cm

Now further finding perimeter of the equilateral triangle

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

Perimeter of equilateral triangle = 3 × 230.94

Perimeter of equilateral triangle = 692.82 cm