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:

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)

**Go To Homepage** **Go To Introduction**
**1) Constructions** **2) Clock Problem** **3) Test Corrections** **4) ASN Explain** **5) Thoughts About Slope** **6) What is Proof?**

**7) Similar Triangles** **8) Homework Corrections** **9) Quads Midpoints** **10) Quads Congruence** **11) Polygons**

**12) Polygons Into Circles** **13) Area and Perimeter** **14) Writing About Grading** **15) Locus** **16) Extra Credit Projects**

**17) Homework Reflections** **18) Students' Overall Reflections** **19) Parents' Evaluate Method** **20) In Conclusion**