Geometry

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.