Shapes and Patterns


Determine a formula in terms of the number of layers that calculates the height of the following configuration of circles .

Number of Layers: 2
Number of Layers: 3


Procedure: Easy

Concept: Easy


This particular Science Fair Project simply demonstrates patterns in geometry. Determining the formula for the layered-circle problem may be trivial. However, the extensions (see below) may offer further ideas for the development of an adequate project. If necessary, brushing up on your Excel (or Open Office) skills for data analysis would be a good idea.


Hypothesize a formula and give reasons for your hypothesis.



  1. Use the example figures in the "Objective" section and determine h for each one.
  2. Record the data in Excel and develop a formula based on the data.
  3. Test the formula for more layers of circles for further verification.
  4. Repeat steps 1-3 if necessary.
  5. Prove your results mathematically. (This page on induction should work.)


Describe the geometric methods you used to derive the formula. What parameters does your result depend on? What parameters does it NOT depend on?


Find ways to generalize your results as much as possible. Can you use information about the radii of the circles and Pythagorean's theorem to determine the height?

If you did not derive a mathematical proof, then be sure to mention the specific parameters that your formula satisfies. Do some research to see if you can find possible applications for this geometric study.


Are there other shapes that you can use? What about different configurations of the same shapes? Of different shapes? Repeating this project in three dimensions may offer more options.