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.