{"id":942,"date":"2022-09-02T02:40:48","date_gmt":"2022-09-01T21:40:48","guid":{"rendered":"https:\/\/rileymath.com\/?p=942"},"modified":"2022-09-02T03:19:04","modified_gmt":"2022-09-01T22:19:04","slug":"geometry-unit-polygons-part-2","status":"publish","type":"post","link":"https:\/\/rileymath.com\/index.php\/2022\/09\/02\/geometry-unit-polygons-part-2\/","title":{"rendered":"Geometry – Unit Polygons (Part 2)"},"content":{"rendered":"\t\t
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\"\"<\/figure><\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Welcome Back to our exploration on the areas of Unit Polygons.<\/p>\n

\u00a0<\/p>\n

So far, we have discovered the areas and perimeters of regular polygons with a Circumradius of one from triangles up to many sides.\u00a0\u00a0Here they are again:<\/p>\n

\u00a0<\/p>\n\n\n\n\n\n
\n

Sides<\/p>\n<\/td>\n

\n

3<\/p>\n<\/td>\n

\n

4<\/p>\n<\/td>\n

\n

5<\/p>\n<\/td>\n

\n

6<\/p>\n<\/td>\n

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7<\/p>\n<\/td>\n

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8<\/p>\n<\/td>\n

\n

9<\/p>\n<\/td>\n<\/tr>\n

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Area<\/p>\n<\/td>\n

\n

1.2999<\/p>\n<\/td>\n

\n

2<\/p>\n<\/td>\n

\n

2.3776<\/p>\n<\/td>\n

\n

2.598<\/p>\n<\/td>\n

\n

2.7365<\/p>\n<\/td>\n

\n

2.8284<\/p>\n<\/td>\n

\n

2.8925<\/p>\n<\/td>\n<\/tr>\n

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Perimeter<\/p>\n<\/td>\n

\n

5.1962<\/p>\n<\/td>\n

\n

5.6569<\/p>\n<\/td>\n

\n

5.8779<\/p>\n<\/td>\n

\n

6<\/p>\n<\/td>\n

\n

6.0744<\/p>\n<\/td>\n

\n

6.1229<\/p>\n<\/td>\n

\n

6.1564<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n
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10<\/p>\n<\/td>\n

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15<\/p>\n<\/td>\n

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20<\/p>\n<\/td>\n

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30<\/p>\n<\/td>\n

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50<\/p>\n<\/td>\n

\n

100<\/p>\n<\/td>\n

\n

150<\/p>\n<\/td>\n

\n

\u221e<\/p>\n<\/td>\n<\/tr>\n

\n

2.9389<\/p>\n<\/td>\n

\n

3.0505<\/p>\n<\/td>\n

\n

3.0902<\/p>\n<\/td>\n

\n

3.1187<\/p>\n<\/td>\n

\n

3.1333<\/p>\n<\/td>\n

\n

3.1395<\/p>\n<\/td>\n

\n

3.1407<\/p>\n<\/td>\n

\n

3.1416<\/p>\n<\/td>\n<\/tr>\n

\n

6.1803<\/p>\n<\/td>\n

\n

6.2374<\/p>\n<\/td>\n

\n

6.2574<\/p>\n<\/td>\n

\n

6.2717<\/p>\n<\/td>\n

\n

6.2791<\/p>\n<\/td>\n

\n

6.2827<\/p>\n<\/td>\n

\n

6.2827<\/p>\n<\/td>\n

\n

6.2832<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Questions:<\/p>\n

\u00a0<\/p>\n

\u03c0 is a significant number in mathematics.\u00a0\u00a0What about those other areas?\u00a0\u00a0Such as 2.598 for a hexagon.\u00a0\u00a0Are they analogous to \u03c0\u00a0for a circle?\u00a0\u00a0Could we say\u00a0<\/span>\u00a0\u00a0Is there a real-world or visual representation of this?<\/p>\n

\u00a0<\/p>\n

There is some serious thinking in these questions if the students can get engaged in them.\u00a0\u00a0The teacher should use all their superpowers to get them engaged in this discussion.<\/p>\n

\u00a0<\/p>\n

To begin, we came up with a general equation for the Area of a Unit Polygon as follows:<\/p>\n

\u00a0<\/p>\n

<\/span><\/p>\n

\u00a0<\/p>\n

This is the trigonometric area of a triangle, which exists in most geometry books and all pre-calculus books.\u00a0\u00a0Where\u00a0n<\/span><\/i>\u00a0is the number of sides and\u00a0C<\/span><\/i>\u00a0is the central angle (360\/n<\/span><\/i>).<\/p>\n

\u00a0<\/p>\n

This can be simplified for unit polygons to this equation where n<\/em> is the number of sides.\u00a0\u00a0I let them discover this in groups.<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n

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\"\"<\/figure>

Desmos graph of Areas vs. Number of sides<\/h3>

It is apparent that it converges on 3.14. What is happening below 3?!?!. This is a good example of extraneous solutions. <\/p><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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As expected, the area converges asymptotically on \u03c0,\u00a0\u00a0but it gave us more questions such as:<\/p>\n

\u00a0<\/p>\n

Obviously, the domain of this graph is the set of all integers from 3 to infinity. However, I like to challenge these limits in class.<\/span><\/p>\n

\u00a0<\/span><\/p>\n

Questions:<\/span><\/p>\n

\u00a0<\/span><\/p>\n

\u00a0<\/p>\n

–\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/span><\/span>What about unit polygons with a number of sides that is not a whole number?\u00a0\u00a0Can it still be called regular?\u00a0\u00a0How would you draw such a polygon?<\/p>\n

–\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/span><\/span>What is happening below n=3?\u00a0\u00a0Does this have a significance?\u00a0\u00a0Can they be called polygons if they have less than three sides?<\/p>\n

\u00a0<\/p>\n

Sometimes in a math class the questions can generate more learning than the answers.\u00a0\u00a0Perhaps the answers should not be pursued but should be allowed to remain in their wondering minds.\u00a0\u00a0Our goal is not always to get correct answers but rather to create strong mathematical minds and mathematical mindsets.\u00a0\u00a0If the questions are all answered, the wonder stops.<\/p>\n

\u00a0<\/p>\n

\u00a0<\/p>\n

Let\u2019s stick with our original question.\u00a0\u00a0What is the significance of the ratio of Perimeter over long diagonal for even unit polygons and long radius + apothem for odd unit polygons.<\/p>\n

\u00a0<\/p>\n