Geometry

Squares, circles, cubes, and cylinders. These are some of the few shapes you would have probably heard of. We call this geometry. Geometry is the study of various shapes and angles. 
 

So how does one define geometry? Geometry is a major branch in mathematics. How these shapes and sizes are applied in our real life world will be discussed in this article.
 

People started using geometry as early as 3000 BC. Geometry was used in ancient Indus valley and Babylonia.

1. Euclidean Geometry: Mathematician Euclid wrote a book called elements. This book formed the basis of Euclidean geometry. It deals with lines, curves, points, angles, etc. Euclidean geometry is of two types — Plane Geometry and Solid Geometry. It is commonly used in fields like physics, astronomy, navigation, and architecture.

2. Non-Euclidean Geometry: The axioms given in Non-Euclidean are similar to Euclidean geometry. However, they have some key differences. Non-euclidean geometry was developed when mathematicians made changes to Euclid’s fifth postulate (parallel postulate).

3. Analytical geometry: It is the study of geometry that uses multiple numbers or coordinates. It gives us accurate positioning of points.

4. Differential Geometry: Another branch of Geometry involves the study of spaces and shapes. It is also the connection between geometry and calculus.

5.Projective Geometry: When dealing with the relationships between geometric figures and the images resulting from projecting them onto other surfaces. This is what we call projective geometry.

4. Convex Geometry: It studies shapes that remain inside the line segment joining two points. It also has applications in functional analysis and optimization.

5. Topology: Shapes undergoing continuous transformations, such as twisting or stretching, is what we call topology, although no point should be torn apart. Physics, biology or even computer sciences are some of the few areas where we use topology.
 

When it comes to understanding geometry, a few fundamentals come to mind, which can be divided into 5 main categories:

1. Area Formulas: These are the geometric formulas used to calculate and measure plane figures. Some important area formulas include:

Rectangle: A = L × W (where ‘L’ is the length and ‘W’ is the width)
Square: A =  S² ( ‘S’ is the side of the square)
Triangle: A = ½ (b x h) ( where ‘b’ is the base and ‘h’ is the height of the triangle)
Circle: A = π × r² (where π can be 3.14 or 22/7 and r is the radius of the circle)

2. Perimeter or Circumference Formulas: To calculate the boundary length of a shape, we use the following formulas:

Rectangle: P = 2 × (length + width)
Square: P = 4 × side
Triangle: P = side1 ​+ side2​ + side3​
Circle: C = 2 × π × radius

3. Volume Formula: Volume is the total amount of space occupied within a solid object. Some basic formulas are given below.

Cube: V =  side³
Cuboid: V = length × width × height
Sphere: V = 4​/3 × π × radius³
Cylinder: V = π × radius² × height

4. Pythagorean Theorem: Pythagoras theorem states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the length of the remaining two sides. The Pythagoras formula is expressed as: a² + b² = ^c2. Where ‘a’ and ‘b’ represent the two sides of the right-angled triangle and ‘c’ is the hypotenuse.

5. Trigonometric Ratios: We use trigonometric ratios to relate the angles and sides of a right-angled triangle. Below are few of the important trigonometric ratios that are commonly used:

sin(θ) = opposite​/hypotenuse
cos(θ) = adjacent​/hypotenuse
tan(θ) = opposite​/adjacent



 

We come across several geometric shapes in our daily lives. Here are some of the common 2D and 3D shapes that we use.

2D Shapes

• Circle
• Triangle
• Rectangle
• Square
• Parallelogram
• Trapezoid

3D Shapes

• Cube
• Sphere
• Cylinder
• Cone
• Pyramid
• Tetrahedron

1. Remembering the Formulas: It is always important to know all the basic formulas in geometry while solving any sum. This will give you a very good advantage.

2. Drawing the Diagrams: Practising drawing the diagram first and then solving might help students know more about the shapes , which will help them understand the concept much better.

3. Always label everything: When solving problems, make sure the labeling of sides and angles of the diagrams are done correctly.

4. Take advantage of special geometry tools: Using tools such as a compass, protractor, divider etc… will help you in reaching the results quicker. 

5. Learn to use graph paper: Graph paper is another key element in geometry which will help you in plotting points and calculating units easily.
 

1. Robotics: Designing and coding robots are possibly mainly because of geometric calculations. It helps with actions and spatial awareness..

2. Urban Planning: Engineers use geometry to plan and build our city streets, which help them plan in a better way.

3. Engineering: Many branches of engineering such as civil, mechanical and aerospace depend on geometry for their designs and analysis.

4. Artists and Designers: Artists use geometry to achieve precise angles and proportions in their designs. One example could be statue carving, where the artist uses simple geometric calculations to give the statue its perfect shape.
 

Geometry is a very important and interesting concept that we can learn by applying. Due to how geometry evolved through the years, many things like architecture and modes of transportation are possible. This is one of the main reasons geometry is essential to understand in our day and age. In the world of AI and Robotics, with a strong understanding of geometry we can achieve these future dreams and ideas.