Page 26 - Appplied Mathematics for the Petroleum and Other Industries, 5th Edition

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6

Practical Geometry

OBJECTIVES

Upon completion of chapter 6, the student will be able to—
1. Find the perimeters of rectangles and other parallelograms, trapezoids,
triangles, and polygons.
2. Find the circumferences of circles and ellipses.
3. Find the length of a triangle’s side with the dimensions of two sides given.
4. Find the areas of parallelograms, trapezoids, triangles, regular polygons,
circles, and ellipses.
5. Solve problems involving plane figures by using geometrical formulas.
6. Find the volumes of rectangular solids, cylinders, elliptical solids, cones,
pyramids, frustums of cones and pyramids, and spheres.
7. Find the lateral and total surface areas of some solid figures.
8. Solve problems involving solid figures by using geometrical formulas.
9. Construct geometric figures by using only a compass and a straightedge.
10. Use triangles and a T-square properly for drawing geometric figures.

INTRODUCTION

Geometry deals with the measurements, properties, and relationships of points,
lines, angles, surfaces, and solids. Plane geometry is concerned with plane, or
two-dimensional, figures, such as squares, rectangles, triangles, and circles. Solid
geometry deals with solid, or three-dimensional, objects, such as cubes, pyramids,
cones, and spheres.
To solve advanced geometry problems, you must use deductive reason-
ing—that is, you must apply logical thinking along with special statements,
which are called theorems, to solve problems. A theorem is a proposition or a
formula that can be solved, or proved, by using basic assumptions and declara-
tions called axioms.
An axiom is a statement or an idea that is accepted as true. For example, a
mathematician may say that a statement or formula is axiomatic. By axiomatic,
the mathematician means that the statement cannot be proven but that it is
Petroleum Extension-The University of Texas at Austin
nevertheless accepted as true. An axiomatic equation, for instance, is 2 + 2 = 4.
It is axiomatic because the equation assumes that everyone agrees that figures
we call numbers exist, that we agree what the numbers stand for, and that if we
add two and two, we get four.
To solve complex geometry problems, you must clearly understand the
terms, comprehend the theorems and their related formulas, and be able to
draw conclusions based on given facts. Students of pure geometry spend a great
deal of time proving theorems. That is, a theorem is proposed as true and, using
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