如何找到给定一侧的三角形面积？

三角形是由三个边组成的封闭图形。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。

如何找到给定一侧的三角形面积？

解决方案：

Let us assume a to be the side of the triangle.

If we divide the bottom side in two parts, we get,

By Pythagoras theorem, the altitude can be obtained by,

Perpendicular² + Base² = Hypotenuse²

Perpendicular² + (a/2)² = a²

Perpendicular² = a² – (a/2)²  

Perpendicular² = 3a²/4

Taking square root,

Perpendicular = √3/2a

Now,

Area of triangle = 1/2 x Perpendicular x Base

= 1/2 x √3/2a x a

On solving, we obtain,

Area = a²(√3/4)

示例问题

问题 1. 求边长为 16 厘米的等边三角形招牌的面积，并求画招牌的成本为 ₹2/cm ² ？（假设 √3 = 1.732）

解决方案：

Here we have to find the cost of painting an equilateral triangular signboard.

Given:

Side of equilateral triangular signboard = 16 cm

First finding the area of the equilateral triangular sign board

As we know that

Area of equilateral triangle = √3/4 × a²

Here a = 16 cm

Area of equilateral triangle = √3/4 × 16²

Area of equilateral triangle = √3/4 × 16 × 16

Area of equilateral triangle = 64√3

Now,

Finding the cost of painting

Given

Cost of painting = ₹2 per cm²

Cost of painting = Area of equilateral triangle × 2

Cost of painting = 64√3 × 2

Cost of painting = 128√3

Using √3 = 1.732 (Given)

Cost of painting = ₹221.696

Therefore,

Cost of painting the equilateral triangular signboard is ₹221.696.

问题 2. 求边长为 32 m 的等边三角形公园的面积？

解决方案：

Here we have to find the area of the equilateral triangular park.

Here we are given that,

Side of the triangular park = 32 m

As we know that

Area of equilateral triangle = √3/4 × a²

Area of equilateral triangular park = √3/4 × a²

Here ‘a’ is the side of the equilateral triangular park is 32 m

Area of equilateral triangular park = √3/4 × 32²

Area of equilateral triangular park = √3/4 × 32 × 32

Area of equilateral triangular park = √3 × 8 × 32 

Area of equilateral triangular park = 256√3 m²

Therefore,

The area of the equilateral triangular park is 256√3 m²

问题 3. 求以 ₹20 pr m ²的比率为海拔 10 m、底面 20 m 铺设等边三角形瑜伽场地的成本？

解决方案：

Here we have to find the cost of carpeting equilateral triangular yoga ground

Given:

Altitude = 10 m

Base = 20 m

As we know that

Area of equilateral triangle = 1/2 × base × height

Area of equilateral triangle = 1/2 × 20 × 10

Area of equilateral triangle = 100 m²

Now finding the cost of carpeting

Cost of carpeting yoga ground at ₹20 pr m²

Cost of carpeting yoga ground = Area of yoga ground × ₹20

Cost of carpeting yoga ground = 100 × ₹20

Cost of carpeting yoga ground = ₹2000

Therefore,

The cost of carpeting the yoga ground is ₹2000.

问题 4. 求一个周长为 18 m 的等边三角形的面积？

解决方案：

Here we have to find the area of the equilateral triangle

Given:

Perimeter of equilateral triangle = 18 m

The Perimeter of equilateral triangle = Side + Side + Side

As all the sides of the equilateral triangle are equal

Perimeter of equilateral triangle = 3 × Side

18 = 3 × Side

Side = 18/3

Side = 6 m

Side of the equilateral triangle is 6 m

Further,

Finding the area of the equilateral triangle

Area of equilateral triangle = √3/4 × a²

a =  side of the equilateral triangle

Area of equilateral triangle = √3/4 × 6²

Area of equilateral triangle = √3/4 × 6 × 6

Area of equilateral triangle = 9√3 m²

问题 5. 如果一个等边三角形的边是 10 m，它的面积是 75 m ² ，那么求这个等边三角形的高？

解决方案：

Here we have to find the height of the equilateral triangle

Given:

Side of equilateral triangle = Base of equilateral triangle = 10 m

Area of equilateral triangle = 75 m²

As we know that

Area of equilateral triangle = 1/2 × base × height

75 = 1/2 × 10 × height

75 = 5 × height

Height = 75/5

Height = 15 m

Therefore,

The height of equilateral triangle is 15 m.