Question 67500: Nellie Flowers applied for the postion of gardener for the estate of the retired eccentric mathematics professor Ben Dannett. Professor Dannett proposed the same problem to each gardneing applicant. The first applicant who was clever enough to solve the problem would get the gardening position - with very handsome salary! Professor Dannett asked each applicant to draw the plans fora garden walk from the following description, showing all dimensions: The outer edges of the walk form a regular polygon, with each side measuring 36 meters. The inner edges of the wall are parallel to the outer edges, and they form the same type of regular polygon, with each side measuring 30 meters. Each interior angle of the regular polygon measures 160 degrees.Needles to say, no applicant had solved the professor's conundrum before Nellie. Nellie solved the problem, and she got the job. How many sides do each of the two regular polygons have? Answer by 303795(595) (Show Source):
You can put this solution on YOUR website! Think about the polygon as having a centre and a number of triangles radiating out from the centre. Each of the triangles must be an isosceles triangle with the two identical angles being 80 degrees. The third angle (at the centre) must be 20 degrees. In a circle there are 360 degrees. The central angles in the triangles radiating from the centre must add to make 360 degrees. At 20 degrees for each central angle there must be 18 angles so the polygon has 18 sides.