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Solids

We all know that geometry is the subject that studies about shapes and figures. There are two types of shapes around us - geometrical shapes and non geometrical shapes. Geometrical shapes are the shapes that have predefined properties explained for that particular type of shape. While, a non geometrical shape is not predefined and does not have common properties. The classification of geometrical figures on the basis of dimensions is as follows -

(1) Zero dimensional shape - A point.
(2)
One dimensional shape - A line having a length as its dimension.
(3) Two dimensional shapes - Having length and breadth as two dimensions. For example - square, triangle, rectangle, parallelogram, trapezoid, rhombus, quadrilateral, polygon, circle etc.
(4) Three dimensional shapes - With length, breadth and width (height) as three dimensions. For example - cube, cuboid, cone, cylinder, sphere, pyramid, prism etc.
(5) Higher dimensional shapes - There are few shapes in expressed in dimensions higher than 3, but we usually do not study them in middle-level mathematics.

Solid is one of the major states of matter. It is characterized by structural rigidity and resistance to change of shape or volume. In geometry, the structure that is in three (even higher) dimension, are known as solids or geometric solids. The study of the properties, volume and surface area of solid objects is called as solid geometry. Let us go ahead and focus more on study of geometrical solids.

 

What is a Solid?

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In geometry, there are various types of solids. Solids are in 3D because they have three dimensions: width, depth and height. The bodies which occupy space are called solids such as, cube, sphere, cuboid, cylinder, cone etc..

Properties of Solids

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Solids are classified in terms of their properties. To analyze characteristics and properties of 3-D geometric shapes, count the number of faces, edges, and vertices in various geometric solids.

Facts about Solids:


Geometric Solids Names Solid Figures
Description
Cube Cube Picture Solid whose faces are square. It has 8 vertices and 12 edges.
Cuboid Cuboid Picture Solid whose faces are rectangular. It has 8 vertices and 12 edges.
Sphere Sphere Picture It is a circular object.
Cylinder Cylinder Picture Cylinder has two flat surfaces and one curved surface.
Cone Cone Picture Cone formed by the locus of all straight line segments that join the apex to the base. It has one curved surface and one flat surface.


Solid Shapes

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There are different types of solids that are classified based on their shapes. The different types of solids are as follows:

  • Cube
  • Cylinder
  • Cone
  • Sphere

Cube:

Cube

Volume of a cube = a3 cubic units

Surface area of a cube = 6a2 square units

Cylinder:

Closed Cylinder

Volume of a cylinder = $\pi r^2 h$ cubic units

Surface area of a cylinder = $2 \pi r(r + h)$ square units

Cone:

Cone

Volume of a cone = $\frac{1}{3}$$ \pi r^2 h$ cubic units

Surface area of a cone = $\pi r (r + l)$ square units

Sphere:

Solid Sphere

Volume of a sphere = $\frac{4}{3}$$\pi r^3$ cubic units

Surface area of a sphere = $4 \pi r^2$ square units

Pictures of Solids

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Given below are some pictures of 3D solids.
Geometric Solids Figures

Solved Examples on Solids

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Given below are some of the examples on solids.

Solved Examples

Question 1:

Find the volume and surface area of a solid object whose each side length is equal to 5.3 centimeter.


Solution:

The given solid object is a cube.

The volume of cube = a3 cubic units

= 5.3 * 5.3 * 5.3

= 148.877

The surface area of cube = 6 a2 square units

= 6 * 5.3 * 5.3

= 168.54 

So, the volume and surface area of the given solid object are 148.877 cubic centimeter and 168.54 square centimeter respectively.



Question 2:

Find the volume and surface area of a cylinder with radius, 3.6 centimeter and height, 5.6 centimeter.


Solution:

The volume of cylinder = $\pi r^2 h$

= 3.14 * 3.6 * 3.6 * 5.6

= 227.89 

Volume of cylinder = 227.89 cubic centimeter

The surface area of cylinder = $2 \pi r(r + h)$

= 2 * 3.14 * 3.6(3.6 + 5.6)

= 2 * 3.14 * 3.6 * 9.2

= 207.99

Surface area = 207.99 square centimeter

So, the volume and surface area of the given cylinder are 227.89 cubic centimeter and 207.99 square centimeter.



Question 3:

Find the volume of a cone, with radius 3 cm and height 9 cm.


Solution:

The volume of cone = $\frac{1}{3}$$\pi r^2 h$ cubic units

$\frac{1}{3}$ * 3.14 * 3 * 3 * 9

= 84.78

Thus, the volume of a cone is 84.78 cubic cm.



More topics in Solids
Volume Formula Lateral Area
Cross Section Cube
Hemisphere Cone
Cuboid Cylinder
Sphere Prism
Pyramid Similar Solids
Polyhedron Surface Area and Volume of Combination of Solids
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