Geometry is a branch of mathematics concerned with questions of shape, size, and the relative position of figures. It originated independently in early cultures as a practical way to measure lengths, areas, and volumes. Euclid formalized geometry in his work Elements around 300 BC, which set the standard for over 2000 years. Modern developments include analytic geometry which represents figures using coordinates and functions, and differential geometry which studies surfaces using calculus. Geometry has applications in fields like astronomy, engineering, and physics.

1. Geometry
Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a
branch of mathematics concerned with questions of shape, size, relative position of
figures, and the properties of space. A mathematician who works in the field of
geometry is called a geometer. Geometry arose independently in a number of early
cultures as a body of practical knowledge concerning lengths, areas, and volumes, with
elements of a formal mathematical science emerging in the West as early as Thales
(6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by
Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to
follow.[1]
Archimedes developed ingenious techniques for calculating areas and volumes,
in many ways anticipating modern integral calculus. The field of astronomy, especially
mapping the positions of the stars and planets on the celestial sphere and describing
the relationship between movements of celestial bodies, served as an important source
of geometric problems during the next one and a half millennia. Both geometry and
astronomy were considered in the classical world to be part of the Quadrivium, a subset
of the seven liberal arts considered essential for a free citizen to master.
The introduction of coordinates by René Descartes and the concurrent developments of
algebra marked a new stage for geometry, since geometric figures, such as plane
curves, could now be represented analytically, i.e., with functions and equations. This
played a key role in the emergence of infinitesimal calculus in the 17th century.
Furthermore, the theory of perspective showed that there is more to geometry than just
the metric properties of figures: perspective is the origin of projective geometry. The
subject of geometry was further enriched by the study of intrinsic structure of
geometric objects that originated with Euler and Gauss and led to the creation of
topology and differential geometry.
Geometry is perhaps the most elementary of the sciences that enable man, by purely
intellectual processes, to make predictions (based on observation) about physical world.
The power of geometry, in the sense of accuracy and utility of these deductions, is
impressive, and has been a powerful motivation for the study of logic in geometry.
Geometry is the mathematical study and reasoning behind shapes and planes in the
universe. Geometry compares shapes and structures in two or three dimemsions.
Geometry is the branch of mathematics that deals with the deduction of the properties,
measurement, and relationships of points, lines, angles, and figures in space from their
defining conditions by means of certain assumed properties of space.
The mathematics of the properties, measurement, and relationships of points, lines,
angles, surfaces, and solids.

2. Branches of Geometry:
Euclidean Geometry
Euclidean, or classical, geometry is the most commonly known geometry, and is the
geometry taught most often in schools, especially at the lower levels. Euclid described
this form of geometry in detail in "Elements," which is considered one of the
cornerstones of mathematics. The impact of "Elements" was so big that no other kind of
geometry was used for almost 2,000 years.
Non-Euclidean Geometry
Non-Euclidean geometry is essentially an extension of Euclid's principles of geometry to
three dimensional objects. Non-Euclidean geometry, also called hyperbolic or elliptic
geometry, includes spherical geometry, elliptic geometry and more. This branch of
geometry shows how familiar theorems, such as the sum of the angles of a triangle, are
very different in a three-dimensional space.
Analytic Geometry
Analytic geometry is the study of geometric figures and constructions using a coordinate
system. Lines and curves are represented as set of coordinates, related by a rule of
correspondence which usually is a function or a relation. The most used coordinate
systems are the Cartesian, polar and parametric systems.
Differential Geometry
Differential geometry studies planes, lines and surfaces in a three-dimensional space
using the principles of integral and differential calculus. This branch of geometry focuses
on a variety of problems, such as contact surfaces, geodesics (the shortest path
between two points on the surface of a sphere), complex manifolds and many more.
The application of this branch of geometry ranges from engineering problems to the
calculation of gravitational fields.