LOGOSquares
Ten squares form regular rings either with adjacent or opposite
vertices touching. Calculate the inner and outer radii of the rings
that surround the squares.
Age
16 to 18
Challenge level
Problem
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Consider figures 1 and 2 showing a 'necklace' and a 'floret' of squares. All the squares are regularly spaced around the circles and have side length 1 unit. Calculate, in each case, the inner and outer radii of the rings (or annuli, the singular is annulus) that surround the squares. If you 'merged' the two figures would one annulus lie completely inside the other?
Getting Started
There are 10 squares in each ring so you know the angles. Find some
right angled triangles.
Student Solutions
Rosalind Goudie from Madras College, St Andrew's sent a good solution to this problem.
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Consider the `necklace' with inner and outer radii $r_1$ and $R_1$. We divide the circle up into ten equal sectors as shown in the diagram. Then $r_1 = {\textstyle {1 \over 2}}\cot ({\pi \over 10})$ and, by Pythagoras' Theorem, $$R_1^2 = (r_1+1)^2+({1 \over 2})^2.$$ Thus, for the `necklace', $r_1= 1.539$ and $R_1= 2.588$.
Consider the `floret' with inner and outer radii $r_2$ and $R_2$. Again we divide the circle up into ten equal sectors as shown in the diagram. Then $R_2 = r_2 +\sqrt{2}$, and $$\tan {\pi\over 10} = {\sqrt{2}/2\over \sqrt{2}/2 + r_2} = {1\over 1+r_2\sqrt{2}}.$$ Thus, for the `floret',$r_2 = 1.469$ and $R_2 = 2.883$. Hence the annulus containing the necklace lies inside the annulus of the floret.
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Teachers' Resources
Why do this problem?
A simple trig. calculation involving cot and tan.
Key question
How many squares in each ring? So what are the angles?
Can you find any right angled triangles?