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## The Five Most Important Concepts In Geometry

After writing an article on everyday uses of geometry and another article on real-world applications of geometry principles, my head is spinning with everything I’ve found. When I’m asked what I consider to be the five most important concepts in the subject, it’s “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 of its importance that I have now. Professors who specialize in this subject may not entirely agree with my choices; but I managed to settle for 5 and did so with these everyday uses and real world applications in mind. Some concepts kept repeating themselves, so they are obviously important in 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 the appropriate measurement units: 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. Obviously you don’t need that knowledge, and now you probably won’t. If you must know, send me an e -mail.) 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 them, I have just given some common units of measurement. There are many more, but you get the concept.

**(2) Polygons.** Here I am referring to shapes made of straight lines. The actual definition is more complicated but not necessary for our purposes. Triangles, quadrilaterals and hexagons are the main 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 figure. This is called “area”, which is actually measured with small squares in it, although the actual measurement is, again, found with formulas and labeled in square inches, or ft ^2 (square feet).

The study of polygons is extended in 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 2-D shape is called “area”, the interior of a 3-D shape is called volume and there are, of course, formulas for that as well.

**(3) Circles.** Because circles are not made of straight lines, our ability to measure distance around the space within them 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: Some math purists 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, all sides of a figure being equal, imagine in your mind or draw on a sheet 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. So some people consider a circle to be a regular (all equal) polygon with an infinite number of sides

**(4) Technical. **It’s not a concept in itself, but in each geometry subject, 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 a round object – when folding n 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 rarely grasped to their full potential.

**(5) Conic Sections.** Imagine a pointed ice cream cone. The word “conical” means cone, and conical section means slices of a cone. Slicing the cone in different ways produces cuts of different shapes. Slicing across gives us a circle. Slicing at an angle turns the circle into an oval or ellipse. Tilted in a different way produces a parabola; and if the cone is a dual, a vertical slice 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 focus is on the applications of these figures – parabolic antennas to send beams of light into the sky, hyperbolic antennas to receive signals from space, hyperbolic curves for musical instruments like trumpets and parabolic reflectors around the bulb of a flashlight. There are elliptical pool tables and exercise machines.

There is another concept that I personally consider to be the **most important of all and that is the study of logic**. The ability to use good reasoning skills is so terribly important and is becoming more and 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’s because one of the parties is misinformed about the rules of logic. Without stressing it too much, but misunderstandings can start wars! Logic has to be taught somehow in **all** school year, and it should be a required course for all students. There is, of course, a reason why this has not happened. In reality, our politicians and people in power depend on an ill-informed population. They rely on it to control. An educated population cannot be controlled or manipulated.

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

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