In geometry, calculation of surface area, curved surface area and volume has own importance. There are various three dimensional shapes are there, such as cuboid, cube, cylinder, cone, sphere, hemisphere etc.
Various formulas are used while calculating surface area, curved surface area and volume of different three dimensional shapes.
Detailed formulas are given below:-
Cuboid:-
Surface area:- 2 × (lb + bh + hl)
Volume:- l × b × h
(l= length, b= breadth, h= height)
Cube:-
Surface area:- 6a²
Volume:- a³
(a= edge)
Cylinder:-
Surface area:- 2πr (r + h)
Curved surface area:- 2πrh
Volume:- πr²h
(r= radius, h= height)
Cone:-
Surface area:- πr (l + r)
Curved surface area:- πrl
Volume:- ⅓ × πr²h
(r= radius, h= height, l= slant height)
Sphere:-
Surface area:- 4πr²
Volume:- ⁴/³ × πr³
(r= radius)
Hemisphere:-
Surface area:- 3πr²
Curved surface area:- 2πr²
Volume:- ⅔ × πr³
(r= radius)
Some basic features of different geometrical three dimensional (3D) shapes.
Cube:-
(1) a cube is a solid geometrical shape.
(2) Cube consists of six square faces.
(3) Cube consists of eight vertices.
(4) Cube has twelve edges.
(5) Length of all sides of a cube are equal.
Cuboid:-
(1) The faces of cuboids are rectangular in shape.
(2) Cuboid has six faces.
(3) Cuboid has eight vertices.
(4) Cuboid consists of twelve edges.
(5) Cuboids are in 3D shape.
Sphere:-
(1) Sphere is a round shaped three dimensional geometrical shape.
(2) Cuboid does not have vertices and edges.
(3) The distance from the centre of a sphere to any point in the sphere is equal.
Hemisphere:-
(1) Hemisphere is half of a sphere.
(2) Hemisphere is a three dimensional shape.
(3) Hemisphere has no vertices and edges.
(4) Hemisphere consists of one circular base and one curved surface.
Cylinder:-
(1) cylinder has two identical and parallel bases.
(2) Cylinder consists of two flat circular faces and one curved face.
(3) Cylinder has two faces.
(4) Cylinder does not have any vertices.
Cone:-
(1) Cone has one face.
(2) Cone consists of one vertices.
(3) Cone does not have any edges.
(4) Cone is a three dimensional geometrical shape.
Some other geometrical concepts are given below:-
Point
Point is an image or shape with no dimensions or it does not contain length, breadth and height. It is merely a dot on a plain paper. Point signifies exact location of a thing.
Line
The image that is formed when points are endlessly extended in one or both directions; That is called a line. Lines do not have a beginning and an end point.
There are two types of lines.
Straight lines
Curved lines
Straight Line
Straight line
A line that continues straight infinitely in both directions is called a straight line.This line continues to infinity on both sides. It does not have a starting point and an endpoint.
Curved Line
A line that does not go straight, but goes crookedly, is called a curved line. It also continues to infinity. Curved lines do not have a starting point and an end point.
Ray
A straight line that runs from an origin to infinity is called a ray. Ray has a starting point but it does not have an end point.
Line segment
A line with a beginning and an end point is called a line segment.
Application of various mathematical aspects / elements/ theories in practical scenario :-
Algebra :-
Algebra is used in various practical aspects in everyday life as well as in different professions. In financial activities, algebra and algebraic equations are often used to calculate expenses, savings, and interest rates etc . It is also used in business organisations, where cost of production, direct expenses and indirect expenses etc are calculated. In construction and architectural activities, algebra is used to calculate areas, volumes, surface area and other dimensions. Algebraic expressions are utilised to calculate discounts, tax, cost price and selling price of multiple items in small scale and medium scale business units. In computer science also, many algorithms, coding, and data interpretation uses algebraic expressions.Algebra is also used in solving time, travel and distance related calculations. In this way algebra is an important part of mathematics which is utilised to solve a variety of practical problems and calculations in practical world.
Trigonometry :-
Trigonometric formulas and expressions are utilised to calculate angles and distances and it helps to design structures, bridges, and buildings etc. Trigonometry is utilised in communication purposes to determine the shortest road and distance between two points. It is helpful for pilots, sailors, and drivers etc to determine the accurate and shortest path. Trigonometry also helps architectural activities such as, it is used in the process of making precise blueprints while making various constructive structures. Proper calculation through trigonometry ensures safety and stability. Again, in the field of physics, proper interpretation is needed. In this respect, trigonometric formulas are utilised in studying waves, frequency force, speed etc. In astronomy also, trigonometry can be helpful while calculating distances between stars and planets. Trigonometry plays a vital role in creating realistic 3D models, animations, and virtual realities, because it includes proper mathematical calculations. Trigonometric calculations help in the field of measurement of land, their distance, angles etc, which is essential for generating maps and land related data.In this way trigonometry is utilised to make calculations more accurate. It helps to interpret all these data according to their uses.
Probability :-
Probability plays a vital role in various parts of branches of mathematics. It has applicability in different fields or branches. Some of these are as follows. In statistical interpretation and calculation of data, probability provides insight to inferences about sample data collection, sampling of data, interpretation of collected data etc. In statistical analysis, probability is a major tool. In various fields, probability helps in estimating risk. These fields include finance, commerce, economics, management etc. Apart from these fields, in the modern day world, probability calculations are utilised in the field of artificial intelligence. In the commerce and business world, probability assumptions are utilised in manufacturing units to minimise cost and expenses. It helps a business unit to make proper decisions.Probability is an important tool in various areas or fields to interpret data and it leads to a concrete conclusion.
I hope, above given formulas and concepts will help students.
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