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Math Notebook of Ian Beardsley
The regular hexagon and pi
Tessellating equilateral triangles we find we can make a regular hexagon, which also tessellates. Making a regular hexagon
like this we find two sides of an equilateral triangle make radii of the regular hexagon, and the remaining side of the equilateral
triangle makes a side of the regular hexagon. All of the sides of an equilateral triangle being the same, we can conclude
that the regular hexagon has its sides equal in length to its radii. If we inscribe a regular hexagon in a circle, we notice
its perimeter is nearly the same as that of the circle, and its radius is the same as that of the circle. If we consider a
unit regular hexagon, that is, one with side lengths of one, then its perimeter is six, and its radius is one. Its diameter
is therefore two, and six divided by two is three. This is close to the value of pi, clearly, by looking at a regular hexagon
inscribed in a circle.
The sum of the angles in a polygon
Draw a polygon. It need not be regular and can have any number of sides. Draw in the radii. The sum of the angles at the
center is four right angles, or 360 degrees. The sum of the angles of all the triangles formed by the sides of the polygon
and the radii taken together are the number sides, n, of the polygon times two right angles, or 180 degrees. The sum of the
angles of the polygons are that of the triangles minus the angles at its center, or A, the sum of the angles of the polygon
equals n(180 degrees)-360 degrees, or
A=180 degrees(n-2)
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