苍鹭的配方

苍鹭公式是一种非常流行的公式，用于在给出三个边的同时求出三角形的面积。这个公式由“ Heron”在他的书“ Metrica”中给出。我们可以将此公式应用于所有类型的三角形，无论是直角，等边还是等腰。苍鹭的公式是

$A = \sqrt{s(s-a)(s-b)(s-c)}$

Where,

A = Area of Triangle ABC

a, b, c = Lengths of the sides of the triangle

s = semi-perimeter = (a + b + c)/2

苍鹭的配方

苍鹭公式的例子

示例1：计算一个三角形的面积，该三角形的边a，b和c的长度分别为14cm，13cm和15cm?

解决方案：

Given:- a = 14cm

             b = 13cm

             c = 15cm

Firstly, we will determine semi-perimeter(s)

s = (a + b + c)/2

s = (14 + 13 + 15)/2

s = 21cm

$A = \sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{21*(21-14)(21-13)(21-15)}$

$A = \sqrt{21*7*8*6}$

$A=\sqrt{3*7*7*4*4*3}$

$A = 7^2+4^2+3^2$

$A = 84cm^2$

例2：如果三角形的边长分别为11cm和13cm，周长为32cm，则求出三角形的面积?

解决方案：

Let a, b and c be the three sides of the triangle.

a = 11cm

b= 13 cm

c =?

Perimeter = 32cm

As we know, Perimeter equals to the sum of the length of three sides of a triangle.

Perimeter = (a + b + c)

32 = 11 + 13 + c

c = 32 – 24

c= 8cm

Now as we already know the value of perimeter,

s = perimeter/2

s = 32/2

s =16cm

a = 11cm, b = 13cm, c = 8cm, s = 16cm

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{16(16-11)(16-13)(16-8)}$

$A = \sqrt{16*5*3*8}$

$A = \sqrt{8*8*30}$

$A = 8\sqrt{30}cm^2$

表面积

固体物体的表面所占的面积可以称为表面积。 通常，“区域”有两种类型：

1. 总表面积
2. 侧面面积

我们将在本文的后半部分详细讨论这两个方面。

立方体

长度，宽度和高度均相等的三维实体称为立方体。立方体由6个面，12个边和8个顶点组成。

立方体

公式1：立方体的总表面积=六个面的总和

得出立方体表面积的公式：

Consider a cube of length l cm, breadth b cm and height h cm.

Area of face ABCD = Area of face EFGH = (l * b)cm²

Area of face AEHD = Area of face BFGC = (b * h)cm²

Area of face ABFE = Area of face DHGC = (l * h)cm²

As we know, Total Surface Area of Cube = Sum of the all the areas of all it’s six faces

Total Surface Area = 2(l * b) + 2(b * h) + 2(l * h)cm²

= 2(l * b + b * h + h * l)cm²

= 2(lb + bh + hl)cm²

As we know, length, breadth and height of a cube are always equal.

= 2(l * l + l * l + l * l)cm²

= 2 ∗ 3l² cm²

Total Surface Area = 6(Edge)²

例子

找到边长为8cm的立方体的表面积?

解决方案：

As we know, Total surface area of cube = 6(Edge)²

here, Edge = 8

Total Surface Area = 6 ∗ (8)²

Total Surface Area of the cube is 384 cm²

公式2：立方体的侧面表面积=四个面的面积之和(不包括底面和顶面)

Lateral Surface Area Of Cube = Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC

Lateral Surface Area = 2(b * h) + 2(l * h) cm²

As we know, all sides of the cube are equal

Lateral Surface Area of cube = 2(l * l + l * l) cm²

Lateral Surface Area Of cube = 4l² = 4(Edge)² 

例子

找出画一个侧面为6cm的方箱的成本，该方箱从顶部以5cm / sq的速率打开?

解决方案：

Side = 6cm

Lateral Surface Area of box = 4(Edge)²

Lateral Surface Area = 4∗ (6)²

The lateral surface area of box = 144 cm² 

Cost of painting = 5 cm²

Total cost = 144 * 5 = 720 Rupees

长方体

具有长度，宽度和高度的固态三维对象，六个直角放置了六个矩形面。像立方体一样，长方体也包含6个面，12个边和8个顶点。

长方体

公式1：长方体的总表面积=其六个面的总和

得出长方体表面积的公式：

Consider a cuboid of length l cm, breadth b cm and height h cm.

Area of face ABCD = Area of face EFGH = (l * b) $cm^2$

Area of face AEHD = Area of face BFGC = (b * h) $cm^2$

Area of face ABFE = Area of face DHGC = (l * h) $cm^2$

As we know, Total Surface Area of Cuboid = Sum of the all the areas of all it’s six faces

Total Surface Area = 2(l * b) + 2(b * h) + 2(l * h) $cm^2$

Total Surface Area of Cuboid= 2(l * b + b * h + h * l) $cm^2$

Total Surface Area of Cuboid = 2(length * breadth + breadth * height + height * length) $cm^2$

公式2：长方体的侧面表面积=四个面的面积之和(不包括底面和顶面)

Lateral Surface Area of Cuboid = Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC

Lateral Surface Area of Cuboid  = 2(b * h) + 2(l * h) $cm^2$

Lateral Surface Area of Cuboid  = 2(l + b) * h $cm^2$

Lateral Surface Area of Cuboid  = 2(Length + Breadth) * Height $cm^2$

例子

计算长，宽和高分别为16cm，8cm和6cm的粉笔盒的总表面积和横向表面积?

解决方案：

As we know that chalk box is cuboidal in shape.

Given, length = 16cm, breadth = 8cm, height = 6cm  

Total Surface Area = 2(length * breadth + breadth * height + height * length)

Total Surface Area = 2(16 * 8 + 8 * 6 +16 * 6) $cm^2$

Total Surface Area = 2(128 + 48 + 96) $cm^2$

Total Surface Area = 544 $cm^2$

Lateral Surface Area = 2(Length + Breadth) * Height

Lateral Surface Area = 2(16 + 8) * 6 $cm^2$

Lateral Surface Area = 288 $cm^2$

右圆圆柱

由矩形绕其一侧旋转产生的实心形状称为直圆柱。示例：稻草，橡胶管。由两个具有相同高度和不同半径的同轴圆柱体界定的实体称为空心圆柱体。

圆筒

令R和r为空心圆柱体的外半径和内半径，h为高度。

公式1：总表面积=外表面积+内表面积+顶部面积+底部面积

推导右圆柱的表面积的公式：

External Surface Area = $2\pi Rh$

Internal Surface Area = $2\pi rh$

Area of top = $\pi (R^2 - r^2)$

Area of base = $\pi (R^2 - r^2)$

Total Surface Area = $2\pi Rh + 2\pi rh+2*\pi(R^2-r^2)$

Total Surface Area = $2\pi h(R+r) +2\pi(R-r)$

Total Surface Area = $2\pi(R+r)(h+R-r)sq.units$

公式2：弯曲(横向)表面积=外部表面积+内部表面积

Curved Surface Area = $2\pi Rh + 2\pi rh$

Curved Surface Area = $2\pi h(R+r)sq.units$

例子

找到高度为14 cm，内部半径= 2cm和外部半径= 3cm的中空右圆柱的总表面积和弯曲表面积?

解决方案：

Given, Height = 14cm , r = 2cm , r = 3cm

Total Surface Area = $2\pi(R+r)(h+R-r)$

Total Surface Area = $2\pi(3+2)(14+3-2) cm^2$

Total Surface Area = $2\pi*5*15 cm^2$

As we know the value of  $\pi$= 3.14(approx)

Total Surface Area = 471 $cm^2$

Curved Surface Area = $2\pi h(R+r)$

Curved Surface Area = 2*3.14*14*5

Curved Surface Area = 439.82 $cm^2$

右圆锥

具有平底和顶点的三维实体。示例：生日帽

锥体

Height = The length of the line joining a vertex to the centre of the base.

Slant height = The length of the line joining a vertex to any point of the circular edge.

Radius = Radius of the base.

Deriving Formula for the Surface area of Right Circular Cone :

Let there be a cone of radius r, height h and slant height l.

Therefore, Length of Circular edge = $2\pi r$

Area of the plane = $\pi r^2$

公式1：圆锥体的总表面积=扇区的面积+基础的面积

Area of the Sector = 1/2 * (arc length) * (radius)

Area of the Sector = $1/2*2\pi r*l$

Area of the Base = $\pi r^2$

Total Surface Area = $\pi rl+\pi r^2$

Total Surface Area = $\pi r(l+r)$

公式2：圆锥形的(横向)曲面=扇区的面积

Curved Surface Area = $\pi rl$

例子

找到倾斜高度为9cm直径为14cm的圆锥体的总表面积和弯曲表面积吗?

解决方案：

Given,  slant height(l) = 9cm,  radius(r) = diameter /2 = 7cm

Curved Surface Area = $\pi rl$

Curved Surface Area = 3.14 * 7 * 9 $cm^2$

Curved Surface Area = 197.9 $cm^2$

Total Surface Area = Curved Surface Area + Area of the Base

Total Surface Area = 197.9 +  $\pi r^2$

Total Surface Area = 197.9 + 153.9 = 351.83 $cm^2$

领域

三维实体，其所有点都与固定点等距，并且是圆形的。示例：球。

球与半球

Note: The sphere has only total surface area.

Let, there be a sphere of radius r.

Surface Area of Sphere =  $4\pi r^2 sq.units$

半球

通过球体中心的平面将球体分为两个相等的部分。他们每个人都称为半球。

Curved Surface Area Of Hemisphere =  $2\pi r^2 sq.units$

Total Surface Area of Hemisphere = Curved Surface Area + Area of Base

Total Surface Area of hemisphere = $2\pi r^2+ \pi r^2$

Total Surface Area of Hemisphere = $3\pi r^2$

例子

找出半径为6厘米的球体的面积，如果将球体分成两个相等的两半，还求出总表面积和弯曲表面积?

解决方案：

Given,  radius(r) = 6cm

Area of Sphere = $4\pi r^2$

Area of sphere = 4 * 3.14 * 6 * 6 $cm^2$

Area of sphere = 452.38 $cm^2$

We know if we divide a sphere into two equal parts, we eventually get two hemispheres of the same area.

Curved Area of the Hemisphere = $2\pi r^2$

Curved Surface Area = 2 * 3.14 * 6 * 6 = 226.19 $cm^2$

Total Surface Area of Hemisphere =  $3\pi r^2$

Total Surface Area of Hemisphere = 3 * 3.14 * 6 * 6 = 339.2 $cm^2$