Basic Geometry |
Triangles, as the prefix tri suggests, are closed geometrical figures that have three straight sides. Every triangle will, as a result, have three angles as well.
Types of Triangles
Special relationships between the three sides and the three angles allow
us to define types of triangles that are nicer to work with and understand,
at least at a basic starting level.
A. The Right Triangle: Perhaps the most basic and important type of
triangle that you will encounter in Astronomy (and in Physics) is the right
triangle. The right triangle is a triangle that has one 90^{o}
angle. Since the sum of the angles in a triangle must be 180^{o},
this implies that the other two angles in a right triangle must add up to
90^{o}. Many special trigonometric relationships between the sides and
the angles emerge as a result of this special 90^{o} angle. One of
these relations is the so called Pythagorean Theorem. For the right
triangle shown in figure 2, the relation is: a^{2} + b^{2}
= c^{2}
Angles
Angles are basic to our observations of the world. We think about angles
all the time, in our perception of reality, even though we my not be aware
of it at any given instant (an in fact most of us think about angles without
ever being aware of it!). Since angles are measurable quantities we have to
have a way of specifying how to measure angles and what numbers and units
to give them.
One basic way of measuring angles is to start somewhere on the plane geometrical
surface - a good, flat landscape - and say that if we go around one revolution,
completeing a circle, we have covered 360 degrees. Thus, the unit of measurment
of angles will then be degrees and 360 degrees = one revolution. The
degree symbol is usually the superscript ^{o}.
A portion of a revolution will of course be smaller than 360 degrees. For
instance, if we go round 1/4th of a circle, as shown in bottom part of the
figure below., then we cover only 1/4th of 360 degrees, which is equal to
90^{o}.
Area
Since the triangle is a bounded figure one can define an area that it encloses. This might seem complicated to do, if you're just staring at a random picture of a triangle. But, one can look at a very simple triangle first and see how we can find out its area. This will not be shown in detail here; if you're interested in the details see the derivation of area of traingle in the supplementary sections. The result is a simple relationship that actually generalizes to all triangles if one correctly identifies the base and the height. The way this is done is as follows: one simply picks a side of the triangle and calls it the 'base' and then from the edge opposite to that side one drops a prependicular line to the side that is picked as 'base' ... this prependicular line is the height. The following figure may help you imagine how this is done. The area of the triangle is then one-half the base times the height:
Notice that the base and the height both have units
of length and so the area A will have units of length^{2},
as expected.
The circle is a closed geometric figure as shown in the following figure:
C = 2 × p × r
where: C = Circumference, r = radius, and p =
3.1415 ....
Area
Again, as for any closed geometric shape, we can also define an area that
is enclosed by a circle. This is called the area of the circle and
it also depends on the radius of the circle. It is given by:
There are many geometric figures to consider, and in fact we can go in a long divergent path into a full course on geometry on these topics. We will not do that here since this is only a set of quick review pages for the math and geomtery that is useful to know when studying astronomy. So, the final geomtrical figures we will consider are the square and the rectangle.
Squares:
The square is defined to be a closed geomteric figure with opposite sides paralles to each other and adjacent sides perpendicular to each other, and withe constraint that all sides are of the same length! These set of constraints restrict us to the following figure:
The perimeter of the square is simple ... only the sides added together.
The area of the square is also simple ... it's the length of the side squared (or the side multiplied by itself).
Rectangles:
Rectangles are somewhat the same idea as squares except that the sides need not be equal. Since it is a closed geometric figure with opposite sides paralles and adjacent sides perpendicular then it has a longer side, usually called its lengh =L , and a shorter side, often called its width= W.