Geometry

Geometry is a branch of mathematics that studies the size, shape, position, and properties of figures and spaces. It also deals with properties such as angles, volume, area, perimeter and many more physical properties of shapes and figures. Geometry is one of the most crucial branches of math as it has immense real life applications. We come across shapes of figures, terms such as area, volume etc in everyday life.

The word geometry is derived from the Greek word ‘Geometron’. Geo means earth and metron mean measurement. Geometry is one of the most ancient branch of mathematics. The origins of geometry can be traced back to 3000 BCE, Greece. From there onwards, there have constant developments and discoveries in the field of geometry. 

Pythagoras: Gave the Pythagorean equation in 530 BCE, which explains the properties of right-angled triangles.

Euclid: In 300 BCE wrote a Systematic compilation of 465 theorems in geometry, including axioms, constructions, and proofs known as the Euclid’s element.

Archimedes: Discovered the method to find area of circles, volume of cylinders and surface of spheres in 250 BCE.

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

Geometry is such a broad topic in itself that it can be further divided into 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 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 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.

Important Geometry Formulas

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: 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’ is the perpendicular side of the triangle  and ‘b’ represent the other perpendicular side of the right-angled triangle and ‘c’ represents 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

Types of Geometric Figures

Geometrical shapes can be broadly classified in to two categories. 2-D shapes and 3-D shapes. Some of them most common 2-D and 3-D shapes are mentioned below:

2D Shapes

 1. Circle

 2. Triangle

 3. Rectangle

 4. Square

 5. Parallelogram

 6. Trapezoid

3D Shapes

1. Cube

2. Sphere

3. Cylinder

4. Cone

5. Pyramid

6. Tetrahedron

When working on geometry and related problems, 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. Remembering the Formulas: It is always important to remember all the basic formulas in geometry while solving any problem. This helps in achieving solutions faster.

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

3. Always label everything: While solving any problem in geometry that contains a shape, marking the sides is an essential step because it helps to avoid any confusion later on in the question.

4. Take advantage of special geometry tools: Using tools such as a compass, protractor, divider, etc. gives an advantage in measuring properties of shapes. 

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.

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 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.
 

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