表面积和体积

对于您可以看到或触摸的任何对象，可以测量三个维度(长度，宽度和高度)。我们所居住的房屋有某些尺寸。您要查看的矩形显示屏/显示器具有其自身长度的宽度和宽度。对于每个三维几何结构，都将测量表面积和体积。

The area or zone covered by the object’s surface is the surface area of any given object. Whereas the quantity of space available in an object is volume.

表面积

三维对象的外表面所占的面积称为表面积。单位为平方。该区域有两种类型：

1. 总表面积
2. 弯曲表面积/侧面表面积

总表面积

包括基部和弯曲部分的区域对应于整个表面积。它是对象表面所包围的面积的数量。如果表单的底面和曲面弯曲，那么这两个区域的总和就是总面积。

弯曲表面积/侧面表面积

除其中心外，曲面面积仅对应于形状的曲面部分的面积。对于圆锥形等形状，通常称为侧面区域。

体积

体积是特定3D对象中的空间量。物体或物质所占据的空间总量称为体积。它以立方单位测量。

表面积和体积公式

给出的表包含总表面积，弯曲表面积/横向表面积以及各种形状的体积。

Name of the shape	Curved Surface Area	Total Surface Area	Volume
Cuboid	2h(l + b)	2(lb + bh + hl)	l * b * h
Cube	4a2	6a2	a3
Cylinder	2πrh	2πr(r + h)	πr2h
Sphere	4πr2	4πr2	4/3π r3
Cone	πrl	πr(r + l)	1/3π r2h
Hemisphere	2πr2	3πr2	2/3π r3

例子

示例1：将每个体积为512 cm ^3的立方体首尾相连。找到得到的长方体的表面积?

解决方案：

Given,

The Volume (V) of each cube is = 512 cm³

we can now implies that a³ = 512 cm³

∴ The side of the cube, i.e. a = 8 cm

Now, the breadth and length of the resulting cuboid will be 8 cm each while its height will be 16  cm.

So, the surface area of the cuboid (TSA) = 2(lb + bh + lh)

Now, by putting the values, we get,

= 2(8 × 16 + 8 × 8 + 16 × 8) cm²

= (2 × 320) = 640 cm²

Hence, TSA of the cuboid = 640 cm²

示例2：我们有一个直径为14厘米，长度为2厘米的圆柱形蜡烛。熔化形成尺寸为7厘米×11厘米×1厘米的长方体蜡烛。可以获得多少个长方体蜡烛?

解决方案：

Dimensions of the cylindrical Candle:

Radius of cylindrical candle = 14/2 cm = 7 cm

Height/Thichkness=2 cm

Volume of one cylindrical candle = πr²h = π x 7 x 7 x (2) cm³ = 308 cm³ .

Volume of cuboid candle = 7 x 11 x 1 = 77 cm³

Hence, number of Cuboidal candles = Volume of cuboid candle/Volume of one cylindrical candle = 308/77 = 4

Hence we can get 4 Cuboidal shaped candles.

示例3：一个女人想要制作一个球形的黏土玩具球，其半径等于她所戴的手镯的半径。考虑到手镯是圆形的，她还希望手镯的面积等于球体的体积。找出她戴的手镯的半径吗?

解决方案：

Let r be the radius of the bangle as well as the sphere,

We have been given that the volume of the sphere is equal to the area of the bangle:

Hence,

πr² = 4/3 πr³

⇒ r = 3/4

Hence the radius of the bangle is 3/4 units.

实施例4：假定直圆锥的倾斜高度为25cm，其高度为24cm。找到圆锥的曲面区域?

解决方案：

The formula for the curved surface area of the cone is πrl. Where r is the radius of the cone and l is the slant height of the cone.

Here the cone is the Right Circular Cone.

So the radius of the cone would be :

$r= \sqrt{l^2 - h^2}$

=> $r = \sqrt{25^2 - 24^2}$

=> r = 7 cm.

Now calculating the curved surface are:

Required Area = (22/7) * 7 * 25 = 550 cm²

Hence the curved surface area of the cone is 550 cm².