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. 

It also deals with the study of points, lines, shapes, and spaces, mainly concerning their size, shape, position, and other properties. Geometry is based on a few basic concepts:
 

• Point: A point is a precise location in space that has no dimensions, no length, width, or height, therefore making it the most basic building block of geometry.
   
• Line: A line is an infinitely long straight path created by an endless number of points. A line is made up of length but has no width or depth.
   
• Plane: A plane is a perfectly flat two-dimensional surface that extends infinitely in all directions and is the plane in which many geometric figures exist.
   
• Postulate: A postulate is a basic (or assumed) truth in geometry that doesn't require proof. A postulate is the basis for developing additional theorems and proofs.

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.

Geometry is broadly classified into several key branches, each addressing specific aspects of shapes and their properties. These branches facilitate a systematic understanding of geometric concepts and are essential in various practical fields such as architecture, engineering, astronomy, design, and technology. 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.

Theorem Example: 
Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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).

Theorem Example: 
Angle Sum Theorem for Hyperbolic Triangles: The sum of the angles in a hyperbolic triangle is always less than 180 degrees.

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

Theorem Example: 
Distance Formula: The distance between two points (x₁,y₁) and  (x₂,y₂) is given by \(\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 }\)

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

Theorem Example: 
Gauss's Theorema Egregium: The Gaussian curvature of a surface is an intrinsic property preserved under local isometry.

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.

Theorem Example: 
Desargues Theorem: Two triangles are in perspective axially if and only if they are in perspective centrally.

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

Theorem Example: 
Helly’s Theorem: For a family of convex sets in Rⁿ, if every n + 1 of them have a point in common, then all the sets have a common point.

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.

Theorem Example: 
Brouwer Fixed Point Theorem: Any continuous function from a closed disk to itself has at least one fixed point.

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.

Theorem Example: 
Bezout’s Theorem: The number of intersection points of two algebraic curves in the plane is equal to the product of their degrees, counting multiplicities.

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

Theorem Example: 
Erdős Distinct Distances Theorem: For any finite set of points, the number of distinct distances determined by these points has a lower bound proportional to \(n \over \sqrt{log n}\) for n points.

Geometry is an important area of mathematics that focuses on the characteristics and relationship of points, lines, shapes, and solids. The study of these things relies on having a solid understanding of some fundamental ideas that enhance spatial understanding and build important problem-solving skills. These fundamentals can be categorized into five essential areas.

1. Area Formulas: The area measures the amount of surface enclosed within a two-dimensional figure. Here are the most important formulas:

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 = \(1 \over 2\) (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: Perimeter refers to the total distance around a polygon, while circumference applies specifically to circles:

Perimeter of a Rectangle:

Perimeter = 2 × (Length + Width)

Perimeter of a Square: 

Perimeter = 4 × Side

Perimeter of a Triangle: 

Perimeter = Side₁ ​+ Side₂​ + Side_3​

Perimeter/Circumference of a Circle: 

Circumference = 2 × π × radius

3. Volume Formula: Volume measures the capacity or space occupied by three-dimensional objects. Some basic formulas are given below:

Volume of a Cube:
Volume = Side³

Volume of a Cuboid:
Volume = Length × Width × Height

Volume of a Sphere:
Volume = \(4 \over 3\) × π × radius³

Volume of a Cylinder:
Volume =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. 

It 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.
'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) = \(Opposite \over Hypotenuse\)
 

cos(θ) = \(Adjacent \over Hypotenuse\)
 

tan(θ) = \(Opposite \over Adjacent \)

cosec (θ) = \( Hypotenuse \over Opposite\)

sec (θ) = \( Hypotenuse \over Adjacent\)

cot (θ) = \(Adjacent \over Opposite\)

Geometrical shapes can be broadly classified into two categories:

1) Two Dimensional (2D) Shapes
2) Three Dimensional (3D) Shapes

Some of the most common 2D and 3D shapes are mentioned below:
 

Two-Dimensional (2D) Shapes:

Two-dimensional (2D) shapes are flat shapes which have only length and width and are entirely on a plane without any thickness. They are defined by their borders, which can be sides or curves and are measured for perimeter and area. Examples of 2D shapes are:
 

• Circle: A collection of points that are all equal distance away from a center point. It has radius and diameter.
   
• Triangle: A polygon with three sides and three angles. They can be classified based on their sides lengths or angles.
   
• Rectangle: A quadrilateral with two equal sides and four right angles.
   
• Square: A special rectangle that has all four sides equal length.
   
• Parallelogram: A quadrilateral with opposite sides that are parallel and equal in length.
   
• Trapezoid (Trapezium): A quadrilateral that has one set of parallel sides.
   

Three-Dimensional (3D) Shapes:

Three-dimensional shapes take up space and have three dimensions: length, width, and height (or depth). Solid figures have volume and surface area, which can be measured. Some key 3D shapes are below: 
 

• Cube: A solid with six square, equal faces. All edges are equal in length. 
   
• Sphere: A solid that is perfectly round solid where every point on the surface is equal distance from the center.
   
• Cylinder: A solid figure that has two parallel circular bases, connected by a curved surface. 
   
• Cone: A solid that has a circular base that tapers up smoothly to a single point at the top. 
   
• Pyramid: A solid that has a polygonal base, and triangular faces that meet at one point. 
   
• Tetrahedron: A special case of a pyramid that has four triangular faces that are all equilateral.

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: