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## The Five Most Important Concepts In Geometry

I have just written an article on the everyday uses of Geometry and another article on real world applications of the principles of Geometry, my head is spinning with everything I have found. To be asked what I consider to be the five most important concepts in the subject is “give me a break”. I spent almost my entire teaching career teaching algebra and avoiding Geometry like the plague, because I didn’t appreciate its importance as much as I do now. Professors specializing in this subject may not fully agree with my choices; but I’ve managed to settle for just 5 and I did it with these everyday uses and real world applications in mind. Some concepts were repeated, so they are obviously important to real life.

**5 most important concepts in 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 measurement: 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 didn’t need that knowledge, and you probably don’t need it now. If you do, email me.) measure 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 a few 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 individual sides, the perimeter, the medians, and so on. Again, these are straight 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 is, again, found with formulas and labeled as square inches, or feet ^ 2 (square feet ).

The study of polygons expands into 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 “inside” of a 2-dimensional figure is called the “area,” the inside of a 3-dimensional figure is called the volume, and of course there are formulas for that as well.

**(3) Circles.** Since circles are not made with straight lines, our ability to measure the distance around the inner space is limited and requires the introduction of a new number: pi. The “perimeter” is actually called the circumference, and both circumference and 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 math purists who think 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. Ready? Now, with all sides of a figure equal, draw in your mind or draw on 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. Thus, some people consider a circle to be a regular polygon (all sides equal) with an infinite number of sides.

**(4) Techniques. **This is not a concept in itself, but in each topic of Geometry you learn techniques to do different things. All 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 a round object; when folding is not an option. There are techniques for dividing a length into thirds or sevenths that would be extremely difficult with manual measurement. All of these techniques are practical applications that are covered in Geometry, but are rarely understood to their full potential.

**(5) Conical sections.** Imagine a spiky ice cream cone. The word “conic” means cone, and conic section means slices of a cone. Cutting the cone in different ways produces different shaped cuts. Cutting straight gives us a circle. Cutting an angle turns the circle into an oval or ellipse. A different angle produces a parabola; and if the cone is double, a vertical cut produces the hyperbola. Circles are generally covered in their own chapter and are not taught as a piece of a cone until conic sections are taught.

The main emphasis is on the applications of these figures: parabolic dishes for sending light rays to the sky, hyperbolic dishes for receiving signals from space, hyperbolic curves for musical instruments such as trumpets, and parabolic reflectors around the light bulb in a flashlight There are elliptical pool tables and exercise machines.

There is one more concept that I personally consider **the most important of all and this is the study of logic**. The ability to use good reasoning skills is very important and increasingly so as our lives become more complicated and global. When two people hear the same words, understand the words, but come to totally different conclusions, it is because one party is uninformed about the rules of logic. Not to put too fine a point on it, but misunderstandings can start wars! Logic must be taught somehow **every** school year, and should be a required course for all college students. Of course, there is a reason why this has not happened. In reality, our politicians and people in power depend on an uninformed population. They rely on this for control. An educated population cannot be controlled or manipulated.

Why do you think there are *talk a lot *on improving education, however *so little action*?

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