Thursday, August 9, 2007

Inside or Outside? A Geometry Investigation...

[A special thanks to Mike for correcting the error I made in question 3 below. It has now been updated!]

BACKGROUND
Students in geometry often see problems involving inscribed and circumscribed circles, squares, triangles and other polygons. Questions involving such diagrams often appear on standardized tests and on math contests as well. Although this particular post focuses only on ratios of areas involving squares, equilateral triangles and circles, I am planning a series of investigations which delves far more deeply, requiring students to discover and verify more general relationships for polygons of n sides. Further, as the number of sides of the polygons increase, students will be asked to analyze the ratio of the areas of the circumscribed to the inscribed polygons and consider if they approach a limiting value. Thus this series of activities prepares students for the calculus as well. Students are also introduced to the duality principle, useful for later study in advanced geometry. A strong background in geometry is needed and, at some point, a knowledge of trigonometric ratios is required. This could be a culminating project for a marking period or the year. I hope you enjoy this and will save it for the school year or ...

Note: Although it would appear to be more logical to have students determine ratios for the triangles first (n = 3), the diagram shows the squares (n=4) on top because the analysis is somewhat easier.

Note: Although this is couched as an investigation for the classroom, my readers are invited to attempt the questions below and suggest various approaches to finding the ratios. Those experienced in this topic will find each question fairly straightforward, however, consider the bigger picture here! Those whose geometry is rusty will need to review some basic properties of circles and polygons.



FOR THE STUDENT:
1. The diagram at the upper left depicts a circle circumscribed about a square and a second circle inscribed in the square. Verify that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:2.
2. The diagram at the upper right may be thought of as the dual to the first diagram, in that each circle has been replaced by a square and the square by a circle. Show that the ratio of the area of the inscribed figure to the area of the circumscribed figure is invariant, that is, it remains 1:2.
3. The diagram at the lower left depicts a circle circumscribed about an equilateral triangle and a second circle inscribed in the triangle. Show that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:4.
4. Similar to #2 except that the circles and the triangle have now been interchanged. Again, show that the ratio is invariant.

4 comments:

Jackie said...

Printing this now to take to Michigan for the weekend. Can't wait to read the ensuing conversation when I get back!

Mike said...

Dave,

For problem #3, isn't the ratio of the areas 1:4 ?

Mike

Dave Marain said...

thanks, mike!
Very careless error on my part (not to mention that the diagram certainly should have made that clear!). It's definitely 1:4 and I corrected it. Shows you that I'm actually writing these and not copying the results from another source!! Thanks again - if you see any other errors, let me know. I'm lucky to have such good editors (who are kind)!
Dave

Anonymous said...

I love these. An interesting question I plan to pose in a department meeting on Monday: Given a regular pentagon where s=1, find the length of the side of the largest square you can fit inside the pentagon ("inscribed" may imply all four vertices are on the pentagon--this may be the largest, or it may not be...). Doable, but requires some thinking.