Chapter 12 - Polygons into Circles

Having studied polygons, we now turned our attention to a new (but related) topic. The main idea was to discover which polygons will "fit into" circles. By "fit" we mean, inscribed in the circle, with all vertices of the polygon touching the circle, as shown below. The rules were "draw or construct the polygon, then see of you can draw or construct a circle around it, with all vertices of the polygon on the circle, as in the figure below":

I asked the students if it would make any difference if it were not equilateral. They said no it wouldn't matter and showed me a scalene triangle inscribed in a circle:

Clearly, a square can be inscribed in a circle:

Having seen the square example, at first the students thought the polygon had to be a regular polygon (all sides and angles congruent.) But in their experiments they discovered that this was not the case. As you can see below, a rectangle does fit nicely in a circle, and it is not a "regular" polygon:

And, as a matter of fact, the quadrilateral does not have to be a rectangle, either: the quadrilateral on the left is inscribed in the circle but the one on the right does not "fit" since all four vertices do not lie on the perimeter of the circle. So the next question is "What property, then, is it that makes the one on the left work while the one on the right does not?

The students explored this idea and came to the conclusion that although any rectangle can be arranged so that it can be inscribed in a circle, not all quadrilaterals can be. The next question, of course, was "What property is it that makes this happen?"

After much experimentation in their groups, after much discussion, the students realized that in order for a quadrilateral to fit in a circle like the ones above, the opposite angles must be supplementary! They were very proud of their discovery, particularly because they had figured it out themselves. We could not find this theorem in any textbook, so we they really felt like "real" mathematicians!

In their explanations, written as part of this assignment, the students answered the questions I had posed, with well-written explanations and comments such as the ones that follow:

"When a quadrilateral is forced into a circle, the quad usually gets more specific. By this I mean that the quadrilateral has more properties; for example, a quadrilateral in general has only one property: 4 sides. But when forced to fit inscribed in a circle, the quad gains the property of it's opposite angles being supplementary. That property is not true for all quads." Jeff W.

Jeff went on to say "This relates to the project we did on connecting the midpoints of the quads." (quadsmidpoints.htm) "When the midpoints were connected, each shape made was also more specific than before. By more specific I mean it became a polygon with more properties, like a square has more properties than a rectangle."

Some other questions for the discussion were these: What other polygons can be inscribed in a circle. How about a pentagon? Any pentagon? Can an octagon? What special characteristics would the pentagon or octagon need? I will leave it to the readers to explore this for themselves.

In some years the students experimented with The Geometer's Sketchpad, and in other years they worked with straightedge and compass constructions to explore the polygons. Some years we used what we called our "straw polygons" - these were polygons we created using drinking straws and string. You can see examples of these "learning aids" at the following website:

http://mathforum.org/~sanders/creativegeometry/2.4strawpolygons.htm

Jason, in his journal entry reflecting on his experiences with this project, wrote the following comments: "In our project on quadrilaterals that are inscribed in circles, there were two phases. The first phase we worked on our small groups using the Geometer's Sketchpad. We were able to move and change the different quadrilaterals with relative freedom, which helped us see what changes and what remains the same. For example, when you try to make a trapezoid fit in a circle, you will find that the random trapezoid you start with probably won't fit. When you drag it around a bit, you see that you can make it fit in a circle, but only if you make it symmetrical: it has to be an isosceles trapezoid."

In her reflections on this project, Eileen said: "I thought this project was very interesting, and I learned a lot from it. I thought this was a good project because I learned some new theorems which weren't even in the book! I also liked it because I could visually see things and their relationships, and so it was easier to remember the new 'theorem'. This is good because sometimes when I just read things, I don't always remember them. But since I discovered these things myself, they really stick in my head."

If time allowed, we would explore other quadrilaterals also. The students discovered the following: If a rhombus is inscribed in a circle, it would only fit if the angles were right angles, and then it became a square.

A kite will fit in a circle, if you make it the right height and width, but not all kites will fit, as you can see below.

"The search for truth is more precious than its possession."

Einstein, Albert (1879-1955)