After having written an article on the everyday uses of Geometry and another article on the applications of Geometry principles in the real world, my head is spinning with everything I found. Ask me what I consider the five most important concepts in the subject is “give me a break.” I spent most of my teaching career teaching Algebra and avoiding Geometry like the plague, because I didn’t have the appreciation for its importance that I have now. Professors who specialize in this subject may not fully agree with my choices; but I managed to settle on just 5 and I did it considering those everyday uses and real world applications. Certain concepts kept repeating themselves, so they are obviously important to real life.
The 5 most important concepts of geometry:
(1) Measurement. This concept covers a lot of territory. We measure distances both large, like across a lake, and small, like the diagonal of a small square. For linear (straight line) measurement, we use appropriate units of measure: inches, feet, miles, meters, etc. We also measure the size of angles and use a protractor to measure in degrees or use formulas and measure angles in radians. (Don’t worry if you don’t know what a radian is. You obviously haven’t needed that knowledge, and now you probably don’t. If you need to know, email me.) weight: in ounces, pounds or grams; and we measure capacity: either liquid, like quarts and gallons or liters, or dry with measuring cups. For each of these, I have just given some common units of measurement. There are many others, but you get the concept.
(2) Polygons. Here I am referring to shapes made with straight lines. The actual definition is more complicated but not necessary for our purposes. Triangles, quadrilaterals, and hexagons are prime examples; And with each shape there are properties to learn and extra things to measure: the lengths of individual sides, perimeter, medians, etc. Again, these are straight line measurements, but we use formulas and relationships to determine the measurements. With polygons, we can also measure the space INSIDE the figure. This is called “area”, it’s actually measured with little squares inside, although the actual measurement, again, is found with formulas and labeled as square inches or ft^2 (square feet).
The study of polygons expands to three dimensions, so we have length, width, and thickness. Boxes and books are good examples of two-dimensional rectangles given the third dimension. While the “interior” of a two-dimensional figure is called the “area,” the interior of a three-dimensional figure is called the volume, and of course there are formulas for that too.
(3) Circles. Because circles are not made from straight lines, our ability to measure distance around interior space is limited and requires the introduction of a new number: pi. The “perimeter” is actually called the circumference, and both the circumference and the area have formulas involving the number pi. With circles, we can talk about a radius, a diameter, a tangent line, and various angles.
Note: there are mathematical purists who think that a circle is made up of straight lines. If you picture each of these shapes in your mind as you read the words, you will discover an important pattern. Clever? Now, with all the sides of a figure equal, imagine in your mind or draw on a piece of paper a triangle, a square, a pentagon, a hexagon, an octagon, and a decagon. What do you notice happening? Right! As the number of sides increases, the figure looks more and more circular. Therefore, some people consider a circle to be a regular polygon (all sides equal) with an infinite number of sides.
(4) Technician. This is not a concept in itself, but in each Geometry subject, techniques are learned to do different things. All of these techniques are used in construction/landscape and many other areas as well. There are techniques that allow us in real life to force lines to be parallel or perpendicular, to force corners to be square, and to find the exact center of a circular area or round object, when folding is not an option. There are techniques for dividing a length into thirds or sevenths that would be extremely difficult if measured by hand. All of these techniques are practical applications that are covered in Geometry but are rarely used to their full potential.
(5) Conical sections. Imagine a pointy ice cream cone. The word “conic” means cone and conic section means slices of a cone. Cutting the cone in different ways produces cuts of different shapes. Cutting in a straight line gives us a circle. Cut at an angle turns the circle into an oval or an ellipse. A different angle produces a parabola; and if the cone is double, a vertical cut produces the hyperbola. Circles are usually covered in their own chapter and are not taught as a cone slice until conic sections are taught.
The main emphasis is on the applications of these figures: parabolic dishes to send light rays into the sky, hyperbolic dishes to receive signals from space, hyperbolic curves for musical instruments such as trumpets, and parabolic reflectors around a flashlight bulb. There are elliptical pool tables and exercise machines.
There is one more concept that I personally consider the most important of all and that is the study of logic. The ability to use good reasoning skills is terribly important and is becoming more so as our lives become more complicated and more global. When two people hear the same words, understand the words, but come to totally different conclusions, it is because one of the parties is not informed about the rules of logic. Not to put too fine a point, but misunderstandings can start wars! Logic needs to be taught somehow in each school year, and should be a required course for all college students. There is, of course, a reason why this has not happened. In reality, our politicians and the powerful depend on an uninformed population. They count on this for control. An educated population cannot be controlled or manipulated.
Why do you think there is so much a lot of talk on improving education, goal so little action?