Geometry
Geometry is one of the oldest branches of mathematics that is concerned with the shape, size, angles, and dimensions of objects in our daytoday life. There are two types of shapes in Euclidean Geometry: Two dimensional and Threedimensional shapes. Flat shapes are 2D shapes in plane geometry that include triangles, squares, rectangles, and circles. 3D shapes in solid geometry such as a cube, cuboids, cones, and so on are also known as solids. Fundamental geometry is based on points, lines, and planes, as described in coordinate geometry.
The various types of shapes in geometry help us relate to shapes we see in our daily lives. With the help of geometric principles, we can easily calculate the area, perimeter, and volume of shapes.
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Definition of Geometry
The word Geometry is made up of two Ancient Greek words ‘Geo’ means ‘Earth’ and ‘metron’ means ‘measurement’. Geometry is the study of different varieties of shapes, figures, and sizes. It gives us knowledge about distances, angles, patterns, areas, and volumes of shapes. The principles of geometry depend on points, lines, angles, and planes. 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
 NonEuclidean Geometry(Elliptical Geometry and Hyperbolic Geometry)
 Convex Geometry
 Topology
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 problemsolving. 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. 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
NonEuclidean Geometry
There are two types of NonEuclidean 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. Topology is where this geometry is used. The planar triangle has a total of angles that is less than 180 degrees, depending on the interior curvature of the curved surface.
Convex Geometry
It consists of convex shapes in Euclidean space and uses techniques that involve real analysis. It is used in various applications of optimization and functional analysis.
Topology
It comprises the properties of space that are under continuous mapping. It is used in consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.
Plane 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
 Lines
 Angles
A point is the nodimensional fundamental unit of geometry. Collinear points are the ones that lie on the same line. A line is a straight path on a plane that extends in both directions with no endpoints. 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. Plane geometry consists of lines, circles, and triangles of two dimensions. Plane geometry is another name for twodimensional geometry.
All twodimensional 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
Acute Angle An Angle between 0 to 90 degrees.
Obtuse Angle– An angle more than 90 degrees but less than 180 degrees.
Right Angle– An angle of 90 degrees.
Straight Angle– An angle of 180 degrees is a straight line.
You can find similar topics discussed in depth in the below articles.
Polygon and its types
A figure that is made up of a finite number of straightline segments closing in a loop. The word ‘poly’ means multiple.
The sum of internal angles of a polygon is : (n2) * 180
where n is the number of sides.
Types of Polygon
The types of polygons are:
 Triangles
 Quadrilaterals
 Pentagon
 Hexagon
 Heptagon
 Octagon
 Nonagon
 Decagon
Here is a list of articles where you can find indepth knowledge about the above topic.
 Polygons and their types
 Measures of the Exterior Angles of a Polygon
 Rectangle, Square, Rhombus, Parallelogram
 Some Special Parallelograms
 Parallel lines and a transversal
 Lines parallel to the same line and Angle Sum Property
 Properties of triangles
 Angle Sum Property of a Triangle
 Inequalities in a triangle
 Theorem – Angle opposite to equal sides of an isosceles triangle are equal
 Angle sum property of a quadrilateral
 Types of quadrilateral
 Properties of Parallelograms
 MidPoint Theorem
 Rhombus
 Kite – Quadrilaterals
 Area of 2D Shapes
 Figures on the same base and between the same parallels
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 indepth knowledge about circles.
 Circles and its Related Terms
 Circle Theorems
 Theorem – There is one and only one circle passing through three given noncollinear points
 Theorem – The sum of opposite angles of a cyclic quadrilateral is 180°
 Inscribed Shapes in a Circle
 Basic Construction
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 indepth knowledge about the above topic.
 Construction of Triangles
 Construction of a Quadrilateral
 Euclid’s Definitions, Axioms, and Postulates
 An equivalent version of Euclid’s Fifth postulate
 Euclid’s Division Algorithm
 Properties of Triangles
 Construction of Similar Triangles
 Similar Triangles
 Pythagoras Theorem and its Converse
 Thales’s Theorem
 Criteria for Similarity of Triangles
 Congruence of Triangles
Solid Geometry
Solid geometry is the study of threedimensional 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 threedimensional. Both threedimensional shapes are created by rotating twodimensional shapes. Essential characteristics of 3D forms are :
 Faces
 Edges
 Vertices
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.
The table consists of the number of edges in different 3D shapes :
Solid Shapes  No. of. Edges 

Triangular Prism  9 
Cube  12 
Rectangular prism  12 
Pentagonal Prism  15 
Hexagonal Prism  18 
Triangular Pyramid  6 
Square Pyramid  8 
Pentagonal Pyramid  10 
Hexagonal Pyramid  12 
Faces
It is defined as the flat surface enclosed by edges that geometric shapes are made up of. It is 2D figure for all 3D figures.
The list of the number of faces in different shapes is given below:
Solid Shapes  No. of. Faces 

Triangular Prism  5 
Cube  6 
Rectangular prism  6 
Pentagonal Prism  7 
Hexagonal Prism  8 
Triangular Pyramid  4 
Square Pyramid  5 
Pentagonal Pyramid  6 
Hexagonal Pyramid  7 
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 vertices in different solid shapes is given in the table:
Solid Shapes  No. of. Vertices 

Triangular Prism  6 
Cube  8 
Rectangular prism  8 
Pentagonal Prism  10 
Hexagonal Prism  12 
Triangular Pyramid  4 
Square Pyramid  5 
Pentagonal Pyramid  6 
Hexagonal Pyramid  7 
Examine these words in depth for various geometric forms here in the following articles:
 Visualizing Solid Shapes
 Mapping Space Around Us
 Cartesian Coordinate System
 Cartesian Plane
 Coordinate Geometry
 Distance formula
 Section formula
 Midpoint Formula
 Area of a Triangle
 Tangent to a circle
 Tangent at any point of a circle is perpendicular to the radius through the point of contact
 Number of Tangents from a point on a circle
 Lengths of tangents drawn from an external point to a circle are equal
 Division of Line Segment in Given Ratio
 Construction of tangents to a circle
 Perimeter of circular figures, Areas of sector and segment of a circle & Areas of combination of plane figures
ThreeDimensional Geometry
Threedimensional 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 indepth knowledge about threedimensional geometry.
 Coordinate Axes and Coordinate Planes in 3D
 Distance Formula & Section Formula
 Slope of a Straight Line
 Introduction to TwoVariable Linear Equations in Straight Lines
 Forms of TwoVariable Linear Equations of a line
 Pointslope Form
 SlopeIntercept Form of Straight Lines
 Writing SlopeIntercept Equations
 Standard Form of a Straight Line
 xintercepts and yintercepts of a Line
 Graphing slopeintercept equations
 Direction Cosines and Direction Ratios of a Line
 Equation of a Line in 3D
 Angle between two lines
 Shortest Distance Between Two Lines in 3D Space
 Points, Lines, and Planes
 Linear Programming
 Graphical Solution of Linear Programming Problems
FAQs on Geometry
Q1: What is Geometry?
Answer:
Geometry is the branch of mathematics that deals with the shape, size, angles, and dimensions of objects in our daytoday life
Q2: What are branches of Geometry?
Answer:
The geometry can be divided into different parts:
 Algebraic Geometry
 Discrete Geometry
 Differential Geometry
 Euclidean Geometry
 Non Euclidean Geometry(Elliptical Geometry and Hyperbolic Geometry)
 Convex Geometry
 Topology
Q3: Why is Geometry important?
Answer:
Geometry is necessary in our daily life to understand the various shapes and quantify them using area and volume.
Q4: What are the basics of Geometry?
Answer:
The basics of geometry are the proper understanding of points, lines, and planes. It then helps in building all other concepts in geometry that are based on these basic concepts.
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