Problem: 12 circles circling a circle

Monday 28 August 2017

Early in the school year, I like to offer my first-year IB maths students - especially HL students - some challenging problems.  I like creating problems of my own that students can solve using mathematics that they would have studied previously. 

Here is my latest creation which is best presented with the dynamic Geogebra applet shown below.  The 12 'outside' circles are all congruent with each being tangent to the larger 'inside' circle and also tangent to the 'outside' circles on either side.

You can challenge your students by asking them to find the ratio of the radius of one of the 'outside' circles, \({r_o}\) , to the radius of the 'inside' circle, \({r_i}\) .  That is, find \(\frac{{{r_o}}}{{{r_i}}}\).  A student (or teacher) should first play with the dynamic Geogebra applet and become convinced that the ratio \(\frac{{{r_o}}}{{{r_i}}}\) is the same regardless the size of the 'inside' circle; in other words, \(\frac{{{r_o}}}{{{r_i}}}\) is a constant.

You can ask for an approximate value of \(\frac{{{r_o}}}{{{r_i}}}\) (e.g. accurate to 3 significant figures) and/or for the exact value of \(\frac{{{r_o}}}{{{r_i}}}\).  Obtaining an exact value is very challenging and there are different ways to express the it.  [see below for access to answer and worked solution]

When I offer problems like this to students, I usually do not require that they solve it for homework.  It is optional.  But, I'm very curious to see which students attempt problems like this - and whether they successfully solve it.  Again, I'm especially interested in how HL students react to problems like this.  It often provides me with some valuable insight into a students' willingness and capability to tackle a genuine problem that is unfamiliar to them - and to get a sense of whether they enjoy it.  I hope they do.  I did.

Here is a link for the Geogebra applet (above) that can be used to access it directly.

Answer: click on 'eye' below to reveal approximate and exact value of \(\frac{{{r_o}}}{{{r_i}}}\).

\(\frac{{{r_o}}}{{{r_i}}} = 3\sqrt 6 - 4\sqrt 3 - 5\sqrt 2 + 7 \approx 0.3491981862...\)

As mentioned, the expression for the exact value can be written in different ways but, I believe, this is the 'simplest'

Here is a two-page PDF file with worked solution & notes:  12 Circles Circling a Circle - Solution

Tags: problem, circle


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