Basic Geometry

Triangles, Circles, and More ...


Contents

  • Triangles

  • Circles

  • Squares, and more ...

    Triangles

    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.



    One important thing to remember about triangles is that the sum of the three angles in a triangle is alway equal to 180 degrees (for definition of degree see subsection below on angles)

    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 90o angle. Since the sum of the angles in a triangle must be 180o, this implies that the other two angles in a right triangle must add up to 90o. Many special trigonometric relationships between the sides and the angles emerge as a result of this special 90o angle. One of these relations is the so called Pythagorean Theorem. For the right triangle shown in figure 2, the relation is: a2 + b2 = c2


    See the Trigonometry section for more - and very IMPORTANT - things to know about right triangles.

    B. Equilateral Triangle: This is a type of triangle in which all 3 sides, and as a result all three angles are equal. Since the sum of the angles in a triangle must be 180o, then this means that for an equilateral triangle each angle is 60o!


    This is rather nice, because eveything about this triangle is determined ... without doing anything you know all three of its angles.

    C. Isoceles Triangle: In this special type of triangle, two sides are equal to each other. One can prove - as you probably did long time ago in your geometry class - that the the angles opposite to these two equal sides are also equal to each other. Thus, let us say that we have a triangle in which two sides are 5.0 cm in length and the other side is - let's say - 2.0 cm in length, as in the figure below. This means that the two angles indicated with an arrow in the figure are equal.


    You will learn how to calculate these angles in the trigonometry section.

    D. Worst Case! Of course, a triangle could just have all 3 sides different in length - as in the first diagram above - in which case there is nothing nice about it! In such cases, one has to find ways of breaking up parts of the triangle or dropping prependiculars to various sides in order to determine anything about the triangle. There is also a very useful tool that is applicable to all traingles in general; it is stated in terms of an equation that is called the the law of cosines , which you can read about in the supplemantary sections.

    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 90o.



    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 length2, as expected.

    Circles

    The circle is a closed geometric figure as shown in the following figure:


    It is defined such that all the points on the circle are at a constant distance from a center. This distance is called the radius of the circle, as indicated in the diagram above.

    Circumference
    The quanitity we call the "circumference" of a geometircal figure is basically the distance one would travel in going around a closed shaped figure once. One can define a circumference for any closed geometric figure. The circumference of a circle is a rather useful quantity, and it depends very simply on the radius of the circle:

    C = 2 × p × r

    where: C = Circumference, r = radius, and p = 3.1415 ....

    You can think of this quantity p as a constant that allows you to define the circumference of a circle. If you're intrested in finding out more about p see the supplementary sections. The value of p also allows us to define other ways of measuing angles (in units called radians.)

    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:

    A = p × r2
    Notice that the value of p comes in again into this formula (as it does for any geometrical measurment of a circle). Also important is to compare this with the circumference of a circle. While the circumferene of a circle depends on r its area depends on r2.
    This observation is significant in terms of units. The circumference will always have the units of distance, while the area will always of units of distance2. You can read more about this in the section on powers.

    Squares, and ...

    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.