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the formula for the sum of the interior angles of a polygon is equal to (n-2)*180.
\n" ); document.write( "the polygon has 12 sides, so the sum of the interior angles of the polygon is equal to (12-2)*180 = 10*180 = 1800 degrees.
\n" ); document.write( "there are 12 angles in the polygon and it is assumed to be a regular polygon (equal angles and equal sides), so each interior angles will be equal to 1800 / 12 = 150 degrees.
\n" ); document.write( "an interior angle and a vertex go together.
\n" ); document.write( "each vertex contains an interior angle that is formed by the sides connecting to that vertex.
\n" ); document.write( "the derivation of that is as follows:
\n" ); document.write( "a straight angle is an angle of 180 degrees.
\n" ); document.write( "the formula stated above of (n-2) * 180 could be re-worded to say (n-2) * a straight angle.
\n" ); document.write( "the formula depends on the fact that the sum of the angles of a triangle is equal to 180 degrees.
\n" ); document.write( "accept that and you can then derive the formula from that.
\n" ); document.write( "look at the picture below and then read the comments below it.
\n" ); document.write( "
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\n" ); document.write( "the sum of the angles in a triangle is equal to 180 degrees.
\n" ); document.write( "each polygon can be divided into the same number of triangles as the sum of the sides of the triangle.
\n" ); document.write( "each of these triangles has a vertex at the center of the polygon.
\n" ); document.write( "a rectangle has 4 sides and can be divided into 4 triangles.
\n" ); document.write( "a hexagon has 6 sides and can be divided into 6 triangle.
\n" ); document.write( "the sum of the angles of the triangle within the polygon is therefore equal to the number of sides * 180.
\n" ); document.write( "if we want to get the sum of the interior angles of the polygon, we have to subtract the sum of the central angles of the polygon.
\n" ); document.write( "the sum of the central angles of the polygon will always be equal to 360 degrees because the number of degrees in a circle is equal to 360 degrees.
\n" ); document.write( "we therefore have to subtract 360 from the sum of the angles of the triangles that form the polygon to get the sum of the interior angles of the polygon.
\n" ); document.write( "this means that our formula becomes:
\n" ); document.write( "n * 180 - 360.
\n" ); document.write( "now, 360 is equal to 2 * 180, so our formula becomes:
\n" ); document.write( "n * 180 - 2 * 180.
\n" ); document.write( "if we factor out the 180, then we are left with (n-2) * 180.
\n" ); document.write( "that's how the formula is derived.
\n" ); document.write( "when you have a dodecagon, then it has 12 sides.
\n" ); document.write( "the formula then becomes:
\n" ); document.write( "(12 - 2) * 180 which becomes 10 * 180.
\n" ); document.write( "here's a link that names the polygons.
\n" ); document.write( "http://mathcentral.uregina.ca/qq/database/QQ.09.96/rosa1.html
\n" ); document.write( "the names are usually derived from the greek names for the number of sides. \n" ); document.write( "