The five most important concepts of geometry

After having written an article about the everyday uses of Geometry and another article about the applications of the principles of Geometry in the real world, my head is spinning with everything I found. Being asked what I consider to be the five most important concepts in the subject is “taking a break.” I spent almost my entire teaching career teaching Algebra and avoiding Geometry like the plague, because I didn’t have the appreciation for its importance that I do now. Professors who specialize in this subject may not fully agree with my choices; but I managed to settle on just 5 and did so 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 do, email me.) weight: in ounces, pounds, or grams; and we measure capacity: either liquid, such as quarts and gallons or liters, or dry with measuring cups. For each of these, I’ve just given some common units of measurement. There are many others, but you get the concept.

(2) Polygons. Here I mean 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 figure there are properties to learn and additional things to measure: the length of the individual sides, the perimeter, the 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 shape. This is called “area”, it’s actually measured with little squares inside, although the actual measurement, again, is found with formulas and is labeled either square inches or feet^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 of straight lines, our ability to measure distance around the interior space is limited and requires the introduction of a new number: pi. The “perimeter” is actually called a circumference, and both circumference and area have formulas that involve the number pi. With circles, we can talk about a radius, a diameter, a tangent line, and various angles.

Note: There are math 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’ll 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 (all sides equal) polygon with an infinite number of sides.

(4) Technical. 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/landscaping 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 discussed in Geometry but are rarely used to their full potential.

(5) Conic sections. Imagine a pointy ice cream cone. The word “conical” means cone, and conical 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 converts the circle to an oval or 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 year of school, 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 much talk on improving education, goal so little action?