## Geometry-Introduction

Geometry is one of the oldest branches of Mathematics that has been used widely since ancient times. Mathematicians have forever been fascinated by the shapes, sizes, and positions of objects like planets, stars and the moon.

A geometer is a mathematician who is specifically concerned with Geometrical studies of shapes and figures.

From the 19th century onwards the field of Geometry has seen wide advancements that have seen several practical applications of geometrical concepts. Geometry is a compulsory subject taught in all schools. Therefore it is important to understand the development of geometrical science over time, its concepts, and practical applications. Interestingly, Geometry is not only applied in Mathematics but also in the field of Physics, Art, Architecture and modern-day AI technology and gaming services.

So whether you are a student or someone looking to explore career options that exploit geometrical concepts, or just someone who is fascinated by shapes and figures, this article has several important pieces of information for you.

### In the following paragraphs, you will be introduced to several important branches of Geometry including Euclidean Geometry, Non-Euclidean Geometries, Analytic Geometry, Projective Geometry, Differential Geometry, and Topology. You will also learn about very basic aspects and components of Geometrical Mathematics. All of the points are explained in a brief and comprehensive manner. The concepts are explained in a simple manner by our experts so that everyone can easily comprehend them.

As a bonus, simple study tips are mentioned towards the end, to help students study the subject of Geometry in the most effective method.

### What is Geometry?

Geometry is a branch of mathematics that deals with the properties, measurement, and relationships of lines, angles, points, coordinates, solids and surfaces, and solids. It is the 4th math course in high school that will help you guide through among other things inclusive of points, planes, parallel lines, lines, angles, quadrilaterals, triangles, squares, similarity, trigonometry, transformations, circles, circumference and area. Learning geometry, you will be able to study the properties of given elements that remain invariant under particular transformations.

### Major Branches of Geometry

1. Euclidean Geometry

In ancient cultures there developed a type of geometry apt to the relationships between lengths, areas, and volumes of physical figures. This geometry gained popularity being codified in Euclid’s elements based upon 10 axioms, or postulates, from which a hundred many theorems were proved by deductive logic.

2. Non-Euclidean Geometries

Several mathematicians substituted alternatives to Euclid’s parallel postulate, which, in its modern form, reads, “Given a line and a point not on the line, it is feasible to construct exactly one line along the given point parallel to the line.”

3. Analytic Geometry

Introduced by the French mathematician René Descartes (1596–1650), this geometry is representative of algebraic equations. This French mathematician only initiated rectangular coordinates to locate points and to allow lines and curves to be delineated with algebraic equations.

4. Projective Geometry

The French mathematician Girard Desargues (1591–1661) initiated projective geometry to enable dealing with those properties of geometric objects that are not revised by projecting their image, or “shadow,” on another surface.

5. Differential Geometry

In connection with practical problems of surveying and geodesy, a German mathematician introduced the domain of differential geometry. Using differential calculus, the innate properties of curves and surfaces are distinguished. For example, he characterized that the intrinsic curvature of a cylinder is just similar to that of a plane, as can be observed by cutting a cylinder through its axis and flattening, but not similar to that of a sphere, which is unable to be flattened without distortion.

6. Topology

Topology, the youngest and most innovative branch of geometry, emphasizes upon the properties of geometric shapes that remain unaltered upon ongoing deformation—stretching, contracting, and folding, but not tearing.

### Geometry Mathematics

Let’s get to know what you will be learning under concepts of geometry:

Lines

Rays, Lines and line segments

Measuring Lines

Parallel, perpendicular, points, and planes

Geometry definitions

The golden ratio

Shapes

Properties and Classification of geometric shapes

Equal parts of shapes

Polygons and Angles with polygons

Curves

Solid geometry (3D shapes)

Angles

Introduction to Angles

Measuring and Constructing Angles

Angles between bisecting lines

Types of Angles

Angles in circle

Triangles

Types of Triangles

Triangle angles

Triangle inequality theorem

Angle bisectors and Perpendicular bisectors

Altitudes, Medians & centroids

Quadrilaterals

Types of Quadrilaterals

Proofs & angles of Quadrilaterals

Coordinate Plane

Coordinate plane: Quadrants on the coordinate plane, quadrant 1 and 4 quadrants

Reflecting points on the coordinate plane

Quadrilaterals and Polygons on the coordinate plane

Area and Perimeter

Count unit squares to find the area

Area of rectangles, trapezoids and Area of parallelograms

Area of triangles

Area of shapes on grids

Area of composite figures

Area and circumference of circles

Advanced area with triangles

Volume and Surface Area

Volume of rectangular prisms

Volume with fractions

Surface area

Surface and volume density

Volume of spheres, cones, and cylinders

Volume and surface area of Solid geometry

Cross sections of 3D objects

Koch snowflake fractal

Heron's formula

### Transformations

Introduction to rigid transformations

Properties and definitions of transformations

Dilations, Translations, Rotations and Reflections of Transformations

Overview of Rigid transformations

Symmetry

Similarity

Introduction and Definition of Similarity

triangle similarity

Solving similar triangles

Angle bisector theorem

Solving with similar and congruent triangles

Solving modeling problems

Congruence

Transformations and congruence

Triangle congruence

Theorems in reference to triangle properties and quadrilateral properties

Working with triangles

Proofs of general theorems that apply triangle congruence

Trigonometry

Introduction to the trigonometric ratios

Special right triangles

Modeling with right triangles

Solving for a side and for an angle in a right triangle using the trigonometric ratios

Trigonometric ratios and similarity

The law of sines and cosines

Sine and cosine of complementary angles

The reciprocal trigonometric ratios

Solving problems on general triangles

Circles

Circle basics

Introduction to radians

Arc measure, Arc length (from degrees), Arc length (from radians)

Sectors

Problem solving on Inscribed angles and Inscribed shapes

Properties of tangents

Area of inscribed triangle

Standard and Expanded equation of a circle

Analytic Geometry

Distance and midpoints

Dividing line segments

Distance on the coordinate plane Problem solving

Parallel and perpendicular lines

Distance between a point and a line challenge

Geometric Constructions

Constructing bisectors of lines and angles

Constructing regular polygons

Constructing inscribed in circles

Constructing incircles and circumcircles

Constructing a line tangent to a circle

Pythagorean Theorem

Introduction and Application of Pythagorean theorem

Pythagorean theorem and distance between points

Pythagorean theorem proofs

Miscellaneous

Worked examples pr problem solving

### How to Study Geometry the Right Way?

Geometry is a very interesting field of Mathematics. When studied the right way, you will not only enjoy learning it but also be able to apply it to various real-time situations. Here is how to make the most of learning geometry:

Read the concepts provided in the textbook a few times before attempting the exercise questions.

Refer to animated videos from different sources to get an added visual aid during studies.

Draw relevant figures for every problem.

Understand various real-life applications of every geometrical concept. This helps to relate to the concepts better and therefore understand them practically. It will also help to apply the concepts to application-based questions easily.

Refer to authentic websites like Vedantu for study material, free classes, videos and personalized tuitions for all the subjects.

## FAQs on Geometry

**1. What is the History Behind Geometry?**

The earliest known unambiguous examples began with devising mathematical rules and methods useful for constructing buildings, surveying land areas, and measuring storage containers. The Greeks around the 6th century BCE collected and delivered this practical knowledge and generalized it in the form of the abstract subject now known as geometry. It is derived from the summation of the Greek words geo (“Earth”) and metron (“measure”) for the measurement of the Earth. Check out the Vedantu website for a detailed and easy explanation.

** 2. What are the Various Geometric Concepts?**

There are majorly 10 geometrical concepts. Geometry guides you through other elements that include:

Points

Planes

parallel lines

Lines

Angles

Triangles

Similarity

Trigonometry

Transformations

Quadrilaterals

Circles and area.

Geometry is an important field of study that has several practical applications. From construction to designing, the understanding of all of these geometrical concepts comes in very handy.

**3. What is the Geometric Concept of Finding the Right Angle?**

Builders and surveyors in ancient times were required to construct right angles onto the field. The technique used by the Egyptians gained them the name “rope pullers” in Greece. This might be because they used a rope for laying out their construction. The surveyor often referred to as a rope stretcher used 3-4-5 triangles and a plummet to carry out his work. These are continued to be used by modern-day surveyors. However, these days there are far more convenient ways that can be used to find a right angle.

**4. Which Ancient Method Helps Finding the Right Angle?**

The simplest method to conduct the trick is to take a rope that is 12 units long, mark knot 3 units from one end and another 5 units from the other end, and then knot the ends together to form a loop. However, the Egyptian scribes generalize them to obtain the Pythagorean Theorem: the square on the line opposite the right angle is equivalent to the sum of the squares on the other two sides.

**5. What are some professions where Geometry is applied extensively?**

There are several professions that use geometrical concepts in their respective fields. Some of these are:

Architecture: a building is essentially a set of various shapes. Architects extensively apply geometrical concepts while designing buildings.

Fashion Designing: this is another profession where geometry is used widely to create dresses of different designs.

Computer-Aided Design engineer: a CAD engineer creates construction plans of different things using geometrical concepts.

Mathematics teacher: Geometrical knowledge is essential for any Mathematics teacher.

Animator: geometry is used extensively in video editing and animations.

Interior designer: interior designers have extensive knowledge of shapes and geometry to design houses as per clients' preferences.

Surveyor: geometry is used by surveyors to draw charts, maps and city plans.

Game developer: this is another developing field where knowledge of geometry is a must.