Geometry by nuhman10


									                         MAT017                               Applied Math II

Geometry is a study of angles, polygons, and solids. We will look briefly at a number of topics,
including angles, various polygons, circles, radian measure, and various solids.


One of the fundamentals of geometry is the angle. An angle is determined by the parts of 2 lines that
have a common endpoint. These are the sides of the angle and the common endpoint is the vertex.
Angles may be designated by a number, a single letter, or by three letters with the center letter being the
vertex, for example Angle 1, Angle A, or Angle BAC.

Three types of angles named for their size are right angle, acute angle, and obtuse angle. A right angle
has exactly 90° while an acute angle has less than 90° and an obtuse angle has more than 90°.

Adjacent angles share a common vertex and one side. Complementary angles add up to 90° and
supplementary angles add up to 180°.


Polygons are shapes that are named for the number of sides. They each have formulas for area and
perimeter that use the designations for sides and height. Sometimes the height (h) of a figure is one of
the sides and sometimes an altitude must be used as the height. An altitude extends from an opposite
corner perpendicular to the base (b). Note: the letters used for base (b) and height (h) are sometimes
denoted as length (l) and width (w).

Quadrilaterals are Polygons with 4 sides. There are various quadrilaterals (4-sided figures). These
include parallelogram, rectangle, square, rhombus, and trapezoid. These quadrilaterals are defined by
features such as right angles, parallel sides, and/or sides of the same length.

Triangles are Polygons with 3 sides. Some special triangles are named by side relationships or angle
components: equilateral, isosceles, scalene, right, acute, and obtuse. The sum of the angles in a
triangle is 180°. The Pythagorean Theorem is a very important tool for determining the unknown
length of a side of a right triangle. Right triangles have one 90° angle which has its vertex designated
by the capital letter C. The other vertices are A and B. The sides opposite these angles are designated by
the corresponding lowercase letters a, b, and c, with c being known as the hypotenuse. The
Pythagorean Theorem uses these side designations to define the length of the hypotenuse c.

                                               c  a2  b2

Similar polygons are the exact same shape, determined by size of the angles, but not the same size,
determined by the length of the sides. Corresponding angles in similar polygons are equal.
Corresponding sides of similar polygons are proportional in length.

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                         MAT017                                Applied Math II

Circles are plane curves consisting of all points at a given distance, called the radius, from a fixed point
in the plane, called the center. Circles also have formulas for perimeter, called the circumference, and
area. These formulas use the radius or diameter of the circle and a value that is designated by the Greek
letter pi (Π). This value is approximately 3.1416 or 22/7 and has its own key on your scientific

Circles have a great many components, or parts, including central angle, inscribed angle, arc, chord,
secant, and tangent. The sum of the measures of all the central angles of any circle is 180°.

                                             Radian Measure
Circles are measured not only in degrees, but also in radians. Just as there are a total of 180° in a circle,
there are also a total of 2Π radians. There is a relationship between degrees and radians which is used
to create the following conversion factors:

                                            rad           180 
                                                  1             1
                                            180           rad

The first is used to convert degrees to radians and the second is used to convert radians to degrees. The
conversions can also be accomplished using these facts:

                                       1°              =       0.01745 rad

                                       1 rad           =       57.3°

                                               Solid Figures
Prisms are solid figures whose sides are parallelograms and whose 2 bases are parallel and congruent
(same size and shape). Formulas include lateral surface area, total surface area, and volume. In solid
geometry the volume formulas often include a capital B which stands for area of the variously shaped

Cylinders are solid figures with a curved lateral surface. The axis of the cylinder is the line segment
between the centers of the bases. The altitude (h) is the perpendicular distance between the bases. When
the axis is the altitude and the bases are circles, the cylinder is called a right circular cylinder.
Formulas include lateral surface area and volume.

Pyramids and Cones have only one base and come up to a point. The sides of a pyramid are triangles
with a common vertex called the apex. Cones have circular bases and a curved lateral surface that
comes to a point called the vertex. Formulas include lateral surface area and volume. If you divide a
pyramid or a cone with a plane, the section between the plane and the base is called a frustum.

Spheres are solid figures formed by a closed curved surface with all points equidistant from the center.
Formulas include surface area and volume.

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