Geometry

Geometry is a branch of mathematics that deals with the study of size, shape and physical properties of figures and spaces. Geometry is one of the most crucial branches of math as it has immense real life applications. It is used in various fields such as architecture, construction, art, engineering, and so on. 

The word geometry is derived from the Greek words, 'geo' meaning earth and 'metron' meaning measure. Geometry is one of the most ancient branches of mathematics. The origins of geometry can be traced back to 3000 BCE, in Greece. From there onwards, there have been constant developments and discoveries in the field of geometry. 

Pythagoras: In 530 BCE, Pythagoras introduced the Pythagorean theorem, which explains the properties of right-angled triangles.

Euclid: In 300 BCE, he wrote a systematic compilation of 465 theorems in geometry. It is known as the Euclid’s Element and includes axioms, constructions, and proofs.

Archimedes: In 250 BCE, he discovered methods to calculate the area of circles, the volume of cylinders, and the surface area of spheres.

Similarly, there have been numerous other discoveries ever since 3000 BCE in the field of geometry. The constant developments and evolution of math has made it advanced enough to solve even complex problems.

A broad topic, geometry can be divided into several subcategories. These categories make it easier to sort shapes and their properties.

1. 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 geometry are similar to those of 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/Coordinate 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 that involves the study of spaces and shapes. It is also the connection between geometry and calculus.

5. Projective Geometry: It is used when dealing with the relationships between geometric figures and the images resulting from projecting them onto other surfaces. This is what we refer to as projective geometry.

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

7. Topology: Shapes undergoing continuous transformations, such as twisting or stretching, are what we call topology, although no point should be torn apart. Physics, biology, or even computer science are among the few areas where we apply topology.

8. Algebraic Geometry: It is a branch of geometry that studies zeros of multivariate polynomials. It includes linear and polynomial algebraic equations used for solving sets of zeros. The application of this type encompasses cryptography, string theory, and other related fields.

9. Discrete Geometry: It is concerned with the relative position of simple geometric objects, such as points, lines, triangles, circles, etc.

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:

Area of a rectangle: 

Area = L × W (where ‘L’ is the length and ‘W’ is the width)

Area of a square: 

Area = S² (‘S’ is the side of the square)

Area of a triangle:

 A = ½ (b x h) (where ‘b’ is the base and ‘h’ is the height of the triangle)

Area of a 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:

Perimeter of a rectangle:

Perimeter = 2 × (length + width)

Perimeter of a square: 

Perimeter = 4 × side

Perimeter of a triangle: 

Perimeter = side1 ​+ side2​ + side3​

Perimeter/Circumference of a circle: 

Circumference = 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: The 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’ is the perpendicular side of the triangle, ‘b’ represents the other perpendicular side of the right-angled triangle, and ‘c’ denotes the hypotenuse.

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

sin(θ) = opposite side ​/ hypotenuse

cos(θ) = adjacent side ​/ hypotenuse

tan(θ) = opposite side / adjacent side

Geometrical shapes can be broadly classified into two categories. 2D shapes and 3D shapes. Some of the most common 2D and 3D shapes are mentioned below:

2D Shapes

•  Circle

•  Triangle

•  Rectangle

•  Square

•  Parallelogram

•  Trapezoid

3D Shapes

• Cube

• Sphere

• Cylinder

• Cone

•  Pyramid

• Tetrahedron

When working on geometry, several key points should be kept in mind to enhance problem-solving efficiency. The following tips and tricks are provided to help you deal with geometry.

1. Remember the Formulas: It is always important to remember all the basic formulas while solving any problem related to geometry. This helps in arriving at solutions faster.

2. Draw: Drawing diagrams when solving geometrical problems gives a better understanding of the shape and its properties.

3. Label Everything: While solving any problem in geometry that contains a shape, marking the sides is an essential step because it helps avoid confusions later on.

4. Use Geometry Tools: Using tools such as compass, protractor, divider, etc., gives an advantage while measuring properties of shapes. 

5. Use Graph Paper: Graph paper is another key element in geometry, which helps you in plotting points and calculating units easily.

Geometry is very essential and commonly used in our daily lives, so let's look at some important areas where it is used:

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

2. Urban Planning: Engineers use geometry to plan and build the city streets. It helps them plan better.

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.
 

From Numbers to Geometry and beyond, you can explore all the important Math topics by selecting from the list below: