Geometry – Definition, Examples, 2D and 3D Shapes & Applications
Geometry is the branch of mathematics that deals with the shapes, angles, dimensions, and sizes of various things that we see in everyday life. Geometry is derived from Ancient Greek words – ‘Geo’ which means ‘Earth’ and ‘metron’ which means ‘measurement’.
Geometry in mathematics plays a crucial role in understanding the physical world around us and has a wide range of applications in various fields, from architecture and engineering to art and physics.
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 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.
In this article, you will learn everything related to Geometry, including geometry, the branches of geometry, the different types of geometry, examples of geometry, and applications of geometry in real life, etc.
Table of Content
 What is Geometry in Maths?
 Geometry Definition
 Branches of Geometry
 Algebraic Geometry
 Discrete Geometry
 Differential Geometry
 Euclidean Geometry
 NonEuclidean Geometry
 Convex Geometry
 Topology
 Plane Geometry
 Important Points in Plane Geometry
 Angles in Geometry
 Polygon and its Types
 Types of Polygon in Geometry
 Circle in Geometry
 Similarity and Congruency in Geometry
 Solid Geometry
 Circle in Geometry
 Similarity and Congruency in Geometry
 Solid Geometry
 ThreeDimensional Geometry
 Geometry Formulas
 Application of Geometry in Real Life
 Geometry Examples
 Practice Problems on Geometry
What is Geometry in Maths?
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.
The word Geometry is made up of two Ancient Greek words ‘Geo’ means ‘Earth’ and ‘metron’ means ‘measurement’.
Geometry Definition
Geometry is a branch of mathematics that studies the properties, measurement, and relationships of points, lines, angles, surfaces, and solids.
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. 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.
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.
Read in Detail: Applications of Topology
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 – A point is the nodimensional 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 twodimensional 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 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°.
 Obtuse Angle– An angle more than 90° but less than 180°.
 Right Angle– An angle of 90°.
 Straight Angle– An angle of 180° 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 in Geometry
The types of polygons are:
Articles Related to Polygons
Here is a list of articles related to Polygons:
 Polygon
 Types of Polygons
 Triangles in Geometry
 Properties of Triangles
 Angle Sum Property of a Triangle
 Triangle Inequality Theorem
 Types of Triangles
 Isosceles Triangle
 Scalene Triangle
 Equiangular Triangle
 Acute Angled Triangle
 Right Angled Triangle
 Obtuse Angled Triangle
 Area of Triangle
 Perimeter of a Triangle
 Types of Quadrilaterals
 Angle Sum Property of a Quadrilateral
 Squares
 Rectangle
 Area of a Rectangle
 Perimeter of a Rectangle
 Parallelogram
 Area of a Parallelogram
 Perimeter of a Parallelogram
 Properties of Parallelograms
 Some Special Parallelograms
 Rhombus
 Trapezium
 Area of a Trapezium
 Perimeter of a Trapezium
 Kites
 Area of a Kite
 Perimeter of a Kite
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.
 Center
 Radius
 Diameter
 Chords
 Tangent
 Secant
 Arc
 Segment
 Sector
 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°
 Lengths of tangents drawn from an external point to a circle are equal
 Inscribed Shapes in a Circle
 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 indepth knowledge about the above topic.
 Construction of Triangles
 Construction of a Quadrilateral
 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
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. 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 indepth knowledge about circles.
 Center
 Radius
 Diameter
 Chords
 Tangent
 Secant
 Arc
 Segment
 Sector
 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°
 Lengths of tangents drawn from an external point to a circle are equal
 Inscribed Shapes in a Circle
 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 indepth knowledge about the above topic.
 Construction of Triangles
 Construction of a Quadrilateral
 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.
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:
Solid Shapes 
No. of. Edges 
No. of. Faces 
No. of. Vertices 

9 
5 
6 

12 
6 
8 

12 
6 
8 

15 
7 
10 

Hexagonal Prism 
18 
8 
12 
6 
4 
4 

8 
5 
5 

10 
6 
6 

Hexagonal Pyramid 
12 
7 
7 
Here is the list of Planar Geometry and Solid Geometry related articles:
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.
 Points, Lines, and Planes
 Coordinate Axes and Coordinate Planes in 3D
 Cartesian Coordinate System
 Cartesian Plane
 Coordinate Geometry
 Distance formula
 Section formula
 Midpoint Formula
 Area of a Triangle in Coordinate Geometry
 Slope of a Straight Line
 Pointslope Form
 SlopeIntercept Form of Straight Lines
 Standard Form of a Straight Line
 X and Y Intercepts
 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
Geometry Formulas
This are some fundamental geometry formulas:
 Rectangle: Area = length × width
 Square: Area = side × side (or side²)
 Triangle: Area = ½ × base × height
 Circle: Area = π × radius²
2. Perimeter/Circumference Formulas
 Rectangle: Perimeter = 2 × (length + width)
 Square: Perimeter = 4 × side
 Triangle: Perimeter = side₁ + side₂ + side₃
 Circle: Circumference = 2 × π × radius
 Cube: Volume = side × side × side (or side³)
 Rectangular Prism: Volume = length × width × height
 Cylinder: Volume = π × radius² × height
 Sphere: Volume = ⁴⁄₃ × π × radius³
For a right triangle with sides \(a\), \(b\), and hypotenuse \(c\): \(a² + b² = c²\).
5. Trigonometric Ratios (for right triangles):
 Sine (sin): sin(θ) = opposite / hypotenuse
 Cosine (cos): cos(θ) = adjacent / hypotenuse
 Tangent (tan): tan(θ) = opposite / adjacent
These are just a few basic formulas; geometry encompasses a wide range of concepts, each with its own set of formulas and principles.
Read More: Geometry Formulas
Application of Geometry in Real Life
 Imagine standing before a towering skyscraper or walking through a charming bridge. The aweinspiring shapes and the strength of these structures owe much to geometry, guiding architects and engineers in creating spaces that are not only safe but also pleasing to the eye.
 Art and design are playgrounds for geometric exploration. Artists manipulate shapes and forms to produce stunning visuals, while designers use geometry to bring balance and harmony to everything from sleek websites to cozy living rooms.
 Next time you lose yourself in a video game or a movie’s CGI wonders, remember that geometry is the secret spell behind those captivating visuals. It helps animate characters and build fantastical worlds that seem nearly as tangible as our own.
 The art of cartography, or mapmaking, transforms the round globe into a flat map with the help of geometry, helping us navigate from point A to point B, whether it’s through bustling city streets or across continents.
 The vastness of space becomes a little more understandable with geometry. It calculates distances to faroff stars and plots the courses of space missions, turning the mysteries of the universe into solvable puzzles.
 In the realm of medicine, geometry’s precise calculations are crucial for technologies like CT scans and MRIs, offering doctors a glimpse inside the human body to diagnose and treat illnesses with remarkable accuracy.
 The convenience of GPS technology, guiding you on your travels or ensuring your online orders arrive at your doorstep, is powered by geometric principles, ensuring accuracy and efficiency in navigation.
 From assembly lines to household helpers, robots rely on geometry to move gracefully and interact with their surroundings, making them indispensable tools in modern manufacturing and daily convenience.
 The next time you dress up or admire a piece of jewelry, consider the geometric principles that influence fashion design, from the symmetry of patterns to the structure of garments.
 Sports aren’t just about physical prowess; they’re also about strategy. Athletes and coaches use geometry to plot gamewinning moves, whether it’s the perfect soccer goal or the ideal swim turn.
Geometry Examples
Example 1: If the equal angles measure 50° in an Isosceles triangle then find the third angle.
Solution:
Let the third angle be x
We know that sum of the three angles of a triangle is 180
⇒ x + 50° + 50° = 180°
⇒ x + 100° = 180°
⇒ x = 180° – 100° = 80°
Hence, the third angle measure 80°
Example 2: If one of the angle measure 70° in a parallelogram, find the rest of all the angles.
Solution:
We know that the sum of the adjacent angles of a parallelogram is 180°. Let the angle adjacent to 70° be x
⇒ 70° + x = 180°
⇒ x = 180° – 70° = 110°
We also know that opposite angles of a parallelogram are equal. Hence, the angle opposite 70° will be 70° and the angle opposite 110° will be 110°
Example 3: If a line of length 3 cm is perpendicular to the Chord of the circle 8 cm then find the radius of the circle.
Solution:
We know that a perpendicular from the center to the chord bisects the chord. Hence, the line from the center will touch the midpoint of the chord such that the length of the line on either side measures 4 cm. Now the perpendicular from the center, half of the chord, and the radius will form a rightangled triangle where the radius will be the hypotenuse of the triangle. Hence, the radius of the circle will be given by using the Pythagoras theorem,
r = √3^{2} + 4^{2} = √25 = 5 cm
Example 4: Find the area of the triangle whose base is 24 cm and height is 12 cm.
Solution:
Area of a triangle is given by 1/2 ⨯ base ⨯ height
Here, base = 24 cm, height = 12 cm
Hence, Area of the triangle is 1/2 ⨯ 24 ⨯ 12 = 144cm^{2}
Example 5: Find the area and circumference of a circle whose radius is 7cm.
Solution:
Given that radius = 7 cm
Circumference of a Circle = 2πr = 2 ⨯ 22/7 ⨯ 7 = 44 cm
Area of Circle = πr^{2} = 22/7 ⨯ 7 ⨯ 7 = 154 cm^{2}
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Practice Problems on Geometry
1. Find the area of a rectangle with a length of 8 cm and a width of 5 cm.
2. A parallelogram has sides of 7 cm and 10 cm. Calculate its perimeter.
3. A quadrilateral has three angles measuring 85°, 90°, and 95°. Find the measure of the fourth angle.
4. Calculate the length of a diagonal of a square with side length 6 cm.
5. Find the area of a rhombus with diagonals measuring 10 cm and 24 cm.
6. Determine the measure of one exterior angle of a regular hexagon.
7. Calculate the volume of a cylinder with a radius of 3 cm and a height of 7 cm.
What is Geometry – FAQs
What is Geometry in maths?
Geometry is the branch of mathematics that deals with the shape, size, angles, and dimensions of objects in our daytoday life.
What are the 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)
 Convex Geometry
 Topology
Why is Geometry important?
Geometry is necessary in our daily life to understand the various shapes and quantify them using area and volume.
What are the basics of Geometry?
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.
What is Euclidean Geometry?
In Euclidean geometry, we study planes and solid figures based on axioms and theorems given by Euclid.
What Is The Difference Between Euclidean and NonEuclidean Geometry?
Euclidean Geometry is the study of the geometry of flat shapes on a plane, while nonEuclidean geometry is the study of the geometry of curved surfaces.
What are the 2 Types of Geometry?
Plane Geometry and Solid Geometry are the 2 types of Geometry. Plane Geometry is about 2D shapes while Solid Geometry is about 3D shapes.
What are the Basics of Geometry?
The Basics of Geometry are the understanding of Points, Lines, Line Segments, and the types of Geometry.
What are the 8 types of geometry?
 Euclidean Geometry: Explores plane and solid figures through axioms and theorems.
 Differential Geometry: Extends calculus principles, crucial in physics for understanding curves and spaces.
 Algebraic Geometry: Focuses on curves and surfaces, utilizing linear and polynomial algebraic equations.
 Discrete Geometry: Analyzes relative positions of basic geometric objects.
 Analytic Geometry: Studies geometric figures and constructions using coordinate systems.
 Riemannian Geometry: Encompasses nonEuclidean geometries, offering diverse geometric perspectives.
 Complex Geometry: Investigates geometric structures based on the complex plane.
 Computational Geometry: Examines properties of explicitly defined algebraic varieties, vital in computational mathematics and computer science.
What is the most common type of geometry?
Euclidean geometry, commonly taught in high schools and featured in precollegiate math contests, is the fundamental geometry type. Also referred to as classical geometry, it focuses on the properties of flat, twodimensional shapes and explores the relationships among points, lines, and angles within a plane.
What is geometry mostly used for?
Geometry is used in many fields, including: Art, Architecture, Engineering, Robotics, Astronomy, Sculptures, Space, Nature, Sports, Machines, Cars.