Fun with Circles
Jeffrey R. Frye
Purpose: The focus of this set of problems is to provide some challenges to students that will allow them to have some fun while attempting to solve the puzzle presented by the construction of the design. These explorations are directed at having some fun by first exploring the how to construct the design, and then discovering the mathematics that are associated with completing these constructions of different kinds of circle designs. Some basic mathematical analysis is presented leaving the more in depth mathematical calculations to the reader or student that has an interest in going beyond the construction itself.
I. Circle Window
There are several items in the first construction that lend themselves to further extension and exploration. The “circle window” offers some interesting challenges for the student to be able to construct this window and not just draw it. By starting with the window and asking the student to explore and find the solution, the challenge is begun. One starting point is to construct three congruent central angles in a circle as shown below.
From the “circle window” design above, it can be seen that three circles are constructed inside the larger circle, and that these circles are tangent to the larger circle, as well as being tangent to each other. How is this accomplished? There are probably several methods that can be used, but the one that is used in this presentation incorporated the properties of the triangle incenter. By constructing three congruent triangles and finding the incenter of each, the three tangent circles inside the larger circle can be constructed. This is one of the hints that will help in constructing the “circle window”. See the construction below.
Now the remainder of the construction requires that arcs be constructed and combined in a way that the design is complete. Click here to explore the “circle window” construction.
II. Bradford Wool Exchange Window
This design incorporates several relationships into the construction. A starting point involves dividing the circle into seven equal arcs and then constructing a heptagon. There are several methods that can be employed to make this construction. Check them out here. This construction used method 1, but the other methods are interesting, too. With the heptagon constructed, the location of the points of tangency for the seven circles with the larger and smaller circles can be located. How are the radii of the small circle and the seven circles determined? The Star of David is then constructed inside the smaller circle. Enjoy working with this one. If you decide that you need help, this will provide some answers and direction. The GSP illustration is a beginning point for this construction; much work is still needed.
III. Double Vesica
This construction uses the relationships found in one of the “special right triangles”, the 30-60-90 right triangle. The intersection of arcs made from this radius relationship, results in the above design. The mathematics has been left to the explorer on this design. Click here to explore using GSP.
IV. Stacking Circles
For this construction, the circles are tangent and the radius of the next lower group of circles is one-half of the radius of the circle above it. By constructing segments that connect the centers of the circles, similar triangles can be constructed. To explore how the tangent circles are constructed and to view the similar triangles, click here.