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Geometric Solids

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Geometric solid is a three dimensional figure which occupies space. Geometric solids have height as its third dimension as an extension to two-dimensional figures. The branch of geometry that deals with geometric solids is known as solid geometry.

What is a Geometric Solid?

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Geometric solid can be defined as the part of space bounded by the sides. Geometric solid is a three-dimensional figure. It occupies some volume in it. The study of geometric solid is called solid geometry or three-dimensional geometry or 3D geometry.

We are surrounded by many three-dimensional figures. All are known as solids. Out of them, few shapes are well defined in geometry which have fixed dimensions like length, breadth, height, radius, etc. These figures are known as geometric solids. Few examples of geometric solids are sphere, pyramid, cone, cube, cuboid and cylinder etc.

Solid Geometric Shapes

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There are various geometrical shapes defined in solid geometry. Few of them are applicable to wide area. These important shapes are shown below:
Cuboid: It has six rectangular faces, 8 vertices and 12 edges.
Cuboid
Cube: It has six square faces, 8 vertices and 12 edges.
Cube
Cylinder: It has two parallel circular faces opposite to a curved surface.
Cylinder
Cone: It has a circular face and a curved surface tapered towards a point.
Cone
Prism: It has two flat polygonal surfaces joined by lateral surfaces.
Prism
Pyramid: It has a flat base and lateral surfaces tapered towards a point.
Pyramid
Sphere:
This is a three-dimensional form of a circle.
Sphere
Hemisphere: A hemisphere is exactly half of a sphere.

Hemisphere

Properties of Geometric Solids

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Few important mathematical properties of geometric solids are listed below:
  • Three-Dimensional Figure: Geometric solids are three-dimensional shapes.
  • Occupies Space: Every geometric solid occupies space.
  • Definite Shape: Every geometrical solid has a definite shape. For example, the shape of a cone is quite different from that of a cylinder. But, all the cones and cylinder have a definite predefined shape.
  • Definite Volume: Volume is the amount of air or space held by the solid. Since every geometric solid is three-dimensional figure, so it contains a definite volume.
  • Fixed Dimensions: Every geometric solid has fixed dimensions. For example, a sphere has radius, a cone has base radius and height, a cuboid has length, breadth and height.
  • Surface Area: Geometric solids have surface area as well. Surface area is referred as the area of outer surface of a solid figure.
  • Possess weight: Geometric solid has some weight because it occupies space in it.
The list of volumes and surface area of few important solid figures are given below:
Solids
Volumes Surface Areas Meaning of Variables
Cuboid  $V=l\ b\ h$  $TSA=2(lb+bh+hl)$
$LSA=2h(l+b)$
l = Length
b = Breadth
h = Height
Cube
$V=a^{3}$
$TSA=6a^{2}$
$LSA=4a^{2}$ 
a = Side
Cylinder
$V=\pi r^{2}h$
$TSA=2\pi r(h+r)$
$LSA=2\pi rh$
r = Radius
h = Height 
Cone
$V$ = $\frac{1}{3}$$\pi r^{2}h$  $TSA=\pi r(l+r)$
$LSA=\pi rl$  
r = Radius
h = Height
l = Slant height
Prism
$V=B\ h$
$TSA=P\ h+2B$
$LSA=P\ h$
B = Area of base
P = Perimeter of base
h = Height
l = Slant height
Pyramid $V$ = $\frac{1}{3}$$B\ h$
$TSA$ = $\frac{1}{2}$$P\ l + B$
$LSA$ = $\frac{1}{2}$$P\ l$
B = Area of base
h = Height
l = Slant height
Sphere
$V$ = $\frac{4}{3}$$\pi r^{3}$  $TSA=LSA=4\pi r^{2}$ r = Radius
Hemisphere
$V$ = $\frac{2}{3}$$\pi r^{3}$  $TSA=3\pi r^{2}$
$LSA=2\pi r^{2}$
r = Radius