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Maths Item of the Month – September 2012 Siders Kites are taken from the following tile: These can be arranged in the following ways: or Would either of these tilings meet-up to form an exact polygon? This activity is taken from the Carom Maths site. For more activities like this please see the Carom Maths website. 1 of 4 JS 01/05/13 © MEI Solution Using the original tile, and the knowledge that the sum of interior angles in a triangle is 180 , we can discern the internal angles in the kites: This allows us to start to examine the arrangements in more detail. For the first arrangement, the important line is the outside line, highlighted below: Arrangement 1 The angle 157.5 is the sum of two interior angles from the kite. As all of the kites are identical, all of the side lengths will be equal, as will all of the internal angles. This implies that any shape generated will be a regular polygon. 2 of 4 JS 01/05/13 © MEI For a regular polygon, with number of sides n , the sum of the interior angles is (n 2) 180 . (n 2) 180 . When equated to this expression, a n valid interior angle of a regular polygon will generate an integer value of n . So to check the Each individual angle must therefore be first configuration we must solve the following equation: Equation: 157.5 (n 2) 180 n Multiplying by n , 157.5n (n 2) 180 Expanding the bracket, 157.5n 180n 360 Rearranging, 360 (180 157.5)n 360 22.5n Dividing by 22.5 , n 16 This means that the angle 157.5 will form a regular polygon with 16 sides. Arrangement 2 This time, the important side is on the inside of the arrangement. Using the same method as before, we find that the angle is, again, 157.5 . As such, the shape generated will still be a 16 sided polygon. 3 of 4 JS 01/05/13 © MEI To illustrate this: Or cut the shapes out and try it yourself: 4 of 4 JS 01/05/13 © MEI