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## The Five Most Important Concepts In Geometry

Having just written an article about everyday uses of Geometry and another article about real world applications of the principles of Geometry, my head is spinning with all I found. Being asked what I consider the five most important concepts in the subject is “giving me pause.” 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 have now. Teachers who specialize in this subject may not totally agree with my choices; but I have managed to settle on just 5 and I did so by considering those everyday uses and real world applications. Certain concepts kept repeating, so they are obviously important to real life.

**5 Most Important Concepts In Geometry:**

**(1) Measurement.** This concept encompasses 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 we use a protractor to measure in degrees or we 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 piece of knowledge, and now you aren’t likely to need it. If you must know, send me an email.) We 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 measure. 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 primary examples; and with each figure there are properties to learn and additional things to measure: the length of individual sides, perimeter, medians, etc. Again, these are straight line measures but we use formulas and relationships to determine the measures. With polygons, we can also measure the space INSIDE the figure. This is called “area,” is measured actually with little squares inside, although the actual measure is, again, found with formulas and labeled as square inches, or ft^2 (feet squared).

The study of polygons gets expanded into three dimensions, so that we have length, width, and thickness. Boxes and books are good examples of 2-dimensional rectangles given the third dimension. While the “inside” of a 2-dimensional figure is called “area,” the inside of a 3-dimensional figure is called volume and there are, of course, formulas for that as well.

**(3) Circles.** Because circles are not made with straight lines, our ability to measure the distance around the space inside is limited and requires the introduction of a new number: pi. The “perimeter” is actually called 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 do think of a circle as being made up of straight lines. If you picture in your mind each of these shapes as you read the words, you will discover an important pattern. Ready? Now, with all sides in a figure being equal, picture 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. Thus, some people consider a circle to be a regular (all equal sides) polygon with an infinite number of sides

**(4) Techniques. **This is not a concept by itself, but in each Geometry topic techniques are learned to do different things. These techniques are all 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 it is not an option. There are techniques for dividing a length into thirds or sevenths that would be extremely difficult with hand measurement. All of these techniques are practical applications that are covered in Geometry but seldom grasped for their full potential.

**(5) Conic Sections.** Picture a pointed ice cream cone. The word “conic” means cone, and conic section means slices of a cone. Slicing the cone in different ways produces cuts of different shapes. Slicing straight across gives us a circle. Slicing on an angle turns the circle into an oval, or an ellipse. Angled a different way produces a parabola; and if the cone is a double, a vertical slice produces the hyperbola. Circles are generally covered in their own chapter and not taught as a slice of a cone until conic sections are taught.

The main emphasis is on the applications of these figures–parabolic dishes for sending beams of light into the sky, hyperbolic dishes for receiving signals from space, hyperbolic curves for musical instruments like 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 that is the study of logic**. The ability to use good reasoning skills is so terribly important and becoming more so as our lives get more complicated and more global. When two people hear the same words, understand the words, but reach totally different conclusions, it is because one of the parties is uninformed about the rules of logic. Not to put too fine a point on it, but misunderstandings can start wars! Logic needs to be taught in some fashion in **every** year of school, and it should be a required course for all college students. There is, of course, a reason why this hasn’t happened. In reality, our politicians, and the power people depend on an uninformed populace. They count on this for control. An educated populace cannot be controlled or manipulated.

Why do you think that there is so *very much talk *about improving education, but *so little action*?

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