Page 27 - Appplied Mathematics for the Petroleum and Other Industries, 5th Edition

148 PRACTICAL GEOMETRY

A axioms, students prove whether it is or is not true. This chapter, however, does
not prove theorems. Rather, it covers practical geometry problems that you may
encounter in the shop or field.
Geometric construction—that is, drawing accurate geometric figures—aids
in visualizing a problem and its solution. So, you should also learn to use a geo-
metric compass, straightedge, and basic drawing tools. (A geometric compass,
unlike the direction finding instrument, is a V-shaped device for drawing circles
or circular arcs.)
90°

B C
PLANE FIGURES
Figure 6.1 Right angle
Lines and angles are the basic elements that make up plane geometric figures.
Examples of plane geometrical figures and a polygon or a circle. A polygon is
any closed figure of three or more straight sides on the same plane. A polygon
has length and width, but no depth. The most common mathematical problems
involving plane figures are determining their perimeters and their areas. The
90° 90° perimeter of a polygon is the total distance of all its sides. The area of a polygon
is the amount of space that it encloses.
Measurements of length and width (or base and altitude, or height) and
angles determine the area and perimeter of a polygon. Polygons include squares,
90° 90°
rectangles, multisided figures such as hexagons and octagons, trapezoids, paral-
lelograms, and triangles. A circle is a closed plane curve, all points of which are
360°
equally distant from a point within called the center. Circle measurements include
circumference, radius, diameter, and area.
Figure 6.2 Circle

Angles and Lines
X An angle is formed when two straight lines, called sides, meet at a point, called
the vertex. For example, in figure 6.1 sides AB and BC meet at point B, forming
the angle ABC. The vertex of the angle is at B. The size of an angle is measured
in degrees (°). One degree is a unit of measurement that is equal to ¹⁄₃₆₀ of a
circle. The angle in figure 6.1 contains 90° and is called a right angle. If a circle
is divided into four equal parts, it contains four right angles of 90° each, or 360°
45° (fig. 6.2). So, a circle contains 360°.
Y Z
An angle that measures less than 90° is an acute angle (fig. 6.3), and an
Figure 6.3 Acute angle angle that measures more than 90° is an obtuse angle (fig. 6.4). The acute angle
in figure 6.3 is a 45° angle. The obtuse angle in figure 6.4 is a 135° angle.
Lines may be straight or curved. Mathematically speaking, lines have
D length but not width and generally lie between two points. Parallel lines are
straight lines in the same plane that do not meet, or intersect, however far ex-
tended. Lines Q and R in figure 6.5 are parallel lines; they are equidistant from
135°
all points on the lines.
Perpendicular lines meet at right angles to each other, like the lines AB and
E F
Petroleum Extension-The University of Texas at Austin
BC in figure 6.1. Put another way, line AB is perpendicular to line BC.
Figure 6.4 Obtuse angle
Parallelograms
Q A parallelogram is a closed figure whose opposite sides are parallel (fig. 6.6).
R If the sides do not meet at right angles, as in figure 6.6, then the figure is a
parallelogram. If, however, the sides meet at right angles—that is, if they are
Figure 6.5 Parallel lines perpendicular to each other—then the parallelogram is a rectangle (fig. 6.7).