Geometry is a branch of mathematics concerned with questions of shape, size, and the properties of space. It arose independently in early cultures and was first put into an axiomatic form by Euclid, whose treatment set a standard for centuries. Later developments included the use of coordinates to represent geometric figures algebraically, contributing to the emergence of calculus, and the study of intrinsic structure leading to new fields like topology and differential geometry. There are several branches of geometry including Euclidean, non-Euclidean, analytic, and differential geometry.
"Application of 3D and 2D geometry" explains the importance of geometry in our lives. Geometry is found everywhere from nature to human made machines. I have tried to inculcate all
its applications.
I hope it helps in providing guidance to those who are aspiring to understand geometry. I have taken help from internet and some books to acquire knowledge.
thank you for clicking my slide.
"Application of 3D and 2D geometry" explains the importance of geometry in our lives. Geometry is found everywhere from nature to human made machines. I have tried to inculcate all
its applications.
I hope it helps in providing guidance to those who are aspiring to understand geometry. I have taken help from internet and some books to acquire knowledge.
thank you for clicking my slide.
A Presentation on the Geometric Bonanza. Hope it will be helpful to students in search of this Topic and even in the topic of Geometry and its Applications. Hope u enjoy it.
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.
This a power point presentation about Euclid, the mathematician and mainly his contributions to Geometry and mathematics. For the full effects, please download it and watch it as a slide show. All comments and suggestions are welcome.
The Presentation explains 'The Father Of Geometry' - "Euclid" with his life history and some of his most influential and remarkable works which contribute to The Modern Mathematics.
Mathematics for Primary School Teachers. Unit 1: Space and ShapeSaide OER Africa
Mathematics for Primary School Teachers has been digitally published by Saide, with the Wits School of Education. It is a revised version of a course originally written for the Bureau for In-service Teacher Development (Bited) at the then Johannesburg College of Education (now Wits School of Education).
The course is for primary school teachers (Foundation and Intermediate Phase) and consists of six content units on the topics of geometry, numeration, operations, fractions, statistics and measurement. Though they do not cover the entire curriculum, the six units cover content from all five mathematics content areas represented in the curriculum.
This unit presents an analytical approach to the study of shapes, including the make-up of shapes, commonalities and differences between shapes and a notation for the naming of shapes.
Junior Primary Mathematics was developed for in-service training of junior primary/foundation phase teachers in South Africa in the late 1990s. However, with the exception of Chapter Three, the topics and approaches will be useful for the training of junior primary mathematics teachers in other African countries. In order to adapt the book, Chapter Three could simply be replaced with a chapter covering the scope and expectations of the national curriculum in the particular country for which it is intended.A pdf version of the resource is also available.
A Presentation on the Geometric Bonanza. Hope it will be helpful to students in search of this Topic and even in the topic of Geometry and its Applications. Hope u enjoy it.
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.
This a power point presentation about Euclid, the mathematician and mainly his contributions to Geometry and mathematics. For the full effects, please download it and watch it as a slide show. All comments and suggestions are welcome.
The Presentation explains 'The Father Of Geometry' - "Euclid" with his life history and some of his most influential and remarkable works which contribute to The Modern Mathematics.
Mathematics for Primary School Teachers. Unit 1: Space and ShapeSaide OER Africa
Mathematics for Primary School Teachers has been digitally published by Saide, with the Wits School of Education. It is a revised version of a course originally written for the Bureau for In-service Teacher Development (Bited) at the then Johannesburg College of Education (now Wits School of Education).
The course is for primary school teachers (Foundation and Intermediate Phase) and consists of six content units on the topics of geometry, numeration, operations, fractions, statistics and measurement. Though they do not cover the entire curriculum, the six units cover content from all five mathematics content areas represented in the curriculum.
This unit presents an analytical approach to the study of shapes, including the make-up of shapes, commonalities and differences between shapes and a notation for the naming of shapes.
Junior Primary Mathematics was developed for in-service training of junior primary/foundation phase teachers in South Africa in the late 1990s. However, with the exception of Chapter Three, the topics and approaches will be useful for the training of junior primary mathematics teachers in other African countries. In order to adapt the book, Chapter Three could simply be replaced with a chapter covering the scope and expectations of the national curriculum in the particular country for which it is intended.A pdf version of the resource is also available.
This learner's module talks about the topic Reasoning. It also includes the definition of Reasoning, Types of Reasoning (Inductive and Deductive Reasoning) and Examples of Reasoning for each type of reasoning.
This Mathematics Learner's module discusses about the basic concepts of Probability and its strategies. It also teaches includes some examples about Probability.
This learner modules talks about the Triangle Inequality. It also talks about the theorems & postulates that supports triangle inequalities in one or two triangles.
with properties of space that are related with distance, shape, .docxfranknwest27899
with properties of space that are related with distance, shape, size, and relative position of figures.
[1]
A mathematician who works in the field of geometry is called a
geometer
.
Until the 19th century, geometry was almost exclusively devoted to
Euclidean geometry
,
[a]
which includes the notions of
point
,
line
,
plane
,
distance
,
angle
,
surface
, and
curve
, as fundamental concepts.
[2]
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is
Gauss
'
Theorema Egregium
(remarkable theorem) that asserts roughly that the
Gaussian curvature
of a surface is independent from any specific
embedding
in an
Euclidean space
. This implies that surfaces can be studied
intrinsically
, that is as stand alone spaces, and has been expanded into the theory of
manifolds
and
Riemannian geometry
.
Later in the 19th century, it appeared that geometries without the
parallel postulate
(
non-Euclidean geometries
) can be developed without introducing any contradiction. The geometry that underlies
general relativity
is a famous application of non-Euclidean geometry.
Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—
differential geometry
,
algebraic geometry
,
computational geometry
,
algebraic topology
,
discrete geometry
(also known as
combinatorial geometry
), etc.—or on the properties of Euclidean spaces that are disregarded—
projective geometry
that consider only alignment of points but not distance and parallelism,
affine geometry
that omits the concept of angle and distance,
finite geometry
that omits
continuity
, etc.
Often developed with the aim to model the physical world, geometry has applications to almost all
sciences
, and also to
art
,
architecture
, and other activities that are related to
graphics
.
[3]
Geometry has also applications to areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental for
Wiles's proof
of
Fermat's Last Theorem
, a problem that was stated in terms of
elementary arithmetic
, and remained unsolved for several centuries.
Contents
1History
2Important concepts in geometry
2.1Axioms
2.2Points
2.3Lines
2.4Planes
2.5Angles
2.6Curves
2.7Surfaces
2.8Manifolds
2.9Length, area, and volume
2.9.1Metrics and measures
2.10Congruence and similarity
2.11Compass and straightedge constructions
2.12Dimension
2.13Symmetry
3Contemporary geometry
3.1Euclidean geometry
3.2Differential geometry
3.2.1Non-Euclidean geometry
3.3Topology
3.4Algebraic geometry
3.5Complex geometry
3.6Discrete geometry
3.7Computational geometry
3.8Geometric group theory
3.9Convex geometry
4Applications
4.1Art
4.2Architecture
4.3Physics
4.4Other fields of mathematics
5See also
5.1Lists
5.2Related topics
5.3Other fields
6Notes
7Sources
8Further .
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.
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.
2. EXAMPLE
Suppose a water tank in the shape of a right circular cylinder is thirty feet long
and eight feet in diameter. How much sheet metal was used in its construction?
What they are asking for here is the surface area of the water tank. The total surface
area of the tank will be the sum of the surface areas of the side (the cylindrical part) and
of the ends. If the diameter is eight feet, then the radius is four feet. The surface area of
each end is given by the area formula for a circle with radius r: A = (pi)r2
. (There are two
end pieces, so I will be multiplying this area by 2 when I find my total-surface-area
formula.) The surface area of the cylinder is the circumference of the circle, multiplied by
the height: A = 2(pi)rh.
Side view of the
cylindrical tank,
showing the radius
"r".
An "exploded"
view of the tank,
showing the three
separate surfaces
whose areas I
need to find.
Then the total surface area of this tank is given by:
2 ×( (pi)r2
) + 2(pi)rh (the two ends, plus the cylinder)
= 2( (pi) (42
) ) + 2(pi) (4)(30)
= 2( (pi) × 16 ) + 240(pi)
= 32(pi) + 240(pi)
= 272(pi)
Since the original dimensions were given in terms of feet, then my area must be in terms
of square feet:
the surface area is 272(pi) square feet.