Introduction to Geometry

Geometry is the branch of mathematics that deals with the shapes, angles, dimensions, and sizes of various things that we see in everyday life. It is mainly divided into plane and solid geometry. In plane geometry, 2d shapes such as triangles, squares, rectangles, and circles are studied. Whereas, in solid geometry, 3d shapes such as cubes, cuboids, cones, etc. are studied.

The basic geometric principles are based on points, lines, and planes explained in coordinate geometry. All the geometrical shapes are based on these geometrical concepts.

Branches of Geometry

The geometry can be divided into different parts:

• Algebraic Geometry
• Discrete Geometry
• Differential Geometry
• Euclidean Geometry
• Non-Euclidean Geometry (Elliptical Geometry and Hyperbolic Geometry)

Algebraic Geometry : This branch of geometry focuses on the zeros of the multivariate polynomial. It consists of linear and polynomial algebraic equations for solving sets of zeros. Applications in this category include string theory and cryptography.

Discrete Geometry: This branch of geometry mainly focuses on the position of simple geometrical objects such as points, lines, triangles, etc. It includes problems based on ordinary continuous spaces that have a combinatorial aspect.

Differential Geometry: It comprises algebraic and calculus techniques for problem-solving. The various problems include problems like general relativity in physics etc.

Euclidean Geometry: In Euclidean geometry, we study planes and solid figures based on axioms and theorems. The fundamental theorems of Euclidean geometry include Points and Lines, Euclid’s Axioms and Postulates, Geometrical Proof, and Euclid’s Fifth Postulate.

Euclidean Geometry

There are two types of shapes in Euclidean Geometry: Two dimensional and Three-dimensional shapes. Flat shapes are 2D shapes that include triangles, squares, rectangles, and circles. 3D shapes such as cubes, cuboids, cones, and so on are also known as solids. Fundamental geometry is based on points, lines, and planes, as described in coordinate geometry.

It has multiple applications in the fields of Computer Science, Mathematics, etc.

The five postulates of Euclidean geometry are as follows:

• A straight line can be drawn from one given point to another.
• The length of a straight line is infinite in both directions.
• Any specified point can serve as the circle's center and any length can serve as the radius.
• All right angles are congruent.
• Any two straight lines that are equal in distance from one another at two points are infinitely parallel.

Some of Euclid's axioms in geometry that are universally accepted are :

• The things that are equal to the same things are equal. If A = C and B = C then A = C
• If equals are added to equals, the wholes are equal. If A = B and C = D, then A + C = B + D
• If equals are subtracted, the remainders are equal.
• The coinciding things are equal t
• The whole is greater than its part. If A > B, then there exists C such that A = B + C.
• The things that are double the same are equal.
• The things that are halves of the same thing are equal 

Non-Euclidean Geometry

There are two types of Non-Euclidean Geometry- Spherical and Hyperbolic Geometry. It is different from Euclidean geometry due to the difference in the principles of angles and parallel lines. 

The study of plane geometry on the sphere is known as spherical geometry. The sum of angles in the triangle is greater than 180°.

A curved surface is referred to as hyperbolic geometry. It is used in Topology.

The planar triangle has a total of angles that is less than 180°, depending on the interior curvature of the curved surface.

Plane Geometry ( 2-D Geometry)

Plane geometry is concerned with the shapes that can be drawn on paper. Euclidean geometry involves the study of plane geometry.

A 2D surface spread infinitely in both directions is referred to as a plane. The basic components of the plane are:

• Points - A point is the no-dimensional fundamental unit of geometry.
• Lines - A line is a straight path on a plane that extends in both directions with no endpoints.
• Angles - Plane geometry consists of lines, circles, and triangles of two dimensions. Plane geometry is another name for two-dimensional geometry. 

Important Points in Plane Geometry

• Collinear points are the ones that lie on the same line.
• A line segment is part of a line that has two endpoints and is finite in length.
• A ray is a line segment that extends indefinitely in one direction. A line has no endpoints.
• Line, line segment, and ray are different from each other.

All two-dimensional figures have only two dimensions: length and width. Plane figures consist of squares, triangles, rectangles, circles, and so on. 

Angles in Geometry

In planar geometry, an angle is formed when two rays intersect, called the sides of the angle, and share a common endpoint known as the vertex of the angle.

There are majorly four types of angles

1. Acute Angle-  An Angle between 0 to 90°.
2. Obtuse Angle- An angle more than 90° but less than 180°.
3. Right Angle- An angle of 90°.
4. Straight Angle- An angle of 180° is a straight line.  

You can find similar topics discussed in depth in the below articles.

1. Lines and Angles
2. Pairs of Angles

Polygon

A figure that is made up of a finite number of straight-line segments closing in a loop. The word 'poly' means multiple. 

The sum of internal angles of a polygon is : (n-2) * 180 

where n is the number of sides.

Types of Polygon

The types of polygons are: 

Articles Related to Polygons

Here is a list of articles related to Polygons:

1. Polygon
2. Triangles in Geometry
3. Properties of Triangles
4. Types of Quadrilaterals
5. Angle Sum Property of a Quadrilateral
6. Parallelogram
7. Some Special Parallelograms

Geometry is one of the oldest branches of mathematics that is concerned with the shape, size, angles, and dimensions of objects in our day-to-day life. Geometry in

Circle in Geometry

A circle is a closed shape. From a fixed point known as the center, all the points of a circle are of the same distance.

Here is a list of articles where you can find in-depth knowledge about circles.

1. Circle Theorems
2. Lengths of tangents drawn from an external point to a circle are equal
3. Inscribed Shapes in a Circle
4. Cyclic Quadrilateral

Similarity and Congruency in Geometry

Similarity: Two figures are considered similar if they have the same shape or equal angle but may not be necessarily of the same size.

Congruence: Two figures are said to be congruent if they are the same shape and size i.e. they are equal in all means.

Here is a list of articles where you can find in-depth knowledge about the above topic.

1. Construction of Triangles
2. Construction of a Quadrilateral
3. Similar Triangles
4. Pythagoras Theorem and its Converse
5. Thales’s Theorem
6. Criteria for Similarity of Triangles
7. Congruence of Triangles

Solid Geometry

Solid geometry is the study of three-dimensional structures such as cubes, prisms, cylinders, and spheres. 3D figures' three dimensions are length, width, and height. However, certain solids do not have faces (e.g. sphere).

The analysis of three dimensions in Euclidean space is known as solid geometry. The structures of our environment are three-dimensional.

Both three-dimensional shapes are created by rotating two-dimensional shapes.

Essential characteristics of 3D forms are :

Edges: An edge is the line segment that joins one vertex to another. It helps in forming the outline of 3D shapes. It means it joins one corner point to another. 

Faces: It is defined as the flat surface enclosed by edges that geometric shapes are made up of. It is a 2D figure for all 3D figures. 

Vertices: A vertex is a point where the edges of the solid figure meet each other. It can be referred to as a point where the adjacent sides of a polygon meet. The vertex is the corner where edges meet. 

The number of edges, faces and vertices in different solid shapes is given in the table:

Here is the list of Planar Geometry and Solid Geometry related articles:

1. Visualizing Solid Shapes
2. Faces, Edges and Vertices

Three-Dimensional Geometry

Three-dimensional geometry studies the geometry of shapes in 3D space in the cartesian planes. Every point in the space is referred by 3 coordinates, (x, y, z) that are real numbers.

Here is a list of articles where you can find in-depth knowledge about three-dimensional geometry.

1. Coordinate Geometry
2. GRE Geometry | Three – Dimensional Figures

Read More: Geometry Formulas