Need help finding how many sides for Geometry

Question

the sum of the measures of the interior angles of a regular polygon is 180 degrees less than three times the sum of the measures of its exterior angles (one at each vertex). How many sides doe the polygon have?

Answer 1

Let's "read and write" exactly what the question says: 

 

The sum of the interior angles of a regular polygon IS:

 

sum interior =

 

 

 

180 degrees less than... (something):

 

sum interior = (something) - 180

 

 

 

three times... (the sum of something):

sum interior = 3(sum of something) - 180

 

 

the sum of the measures of the exterior angles.  Here's where we need to take a diversion:

 

[[The sum of the exterior angles of a regular polygon is 360.  Think of any regular polygon - say a pentagon.  Extend each side out in one direction.  The angle one side of the pentagon makes with the extension of the next side is an extior angle.  It is the supplement of the interior angle.  Now shrink the size of your pentagon down to as small as it gets - virtually a point.  You have 5 lines emanating from this point.  The sum of the 5 angles around that point total 360.]]

 

sum interior = 3(360) - 180

 

sum interior = 900.

 

Now you can use a formula for the sum of the interior angles of a polygon (given below), -- OR  prove to yourself that it is true by constructing the max number of triangles that start from a single vertex (draw lines from one vertiex to all the other vertices.  Each triangle you make contributes 180 degrees to the total sum of the angle of your original polygon):

 

sum interior angles of any polygon  = 180 (n-2), where n is the number of sides of the polygon.

 

So set 900 = 180(n-2) and solve for n.

 

n = (900/180) +2 = 7.

 

Now check.  For a seven sided polygon, the interior angles sum 180(7-2) = 180(5) = 900.

 

Each interior angle of the seven sided figure is (sum of angles)/(number of angles) =  900/7 = 128.57.

 

The exterior angle is the supplement:  180 - 128.57 = 51.43

 

Does 900 = 3( 7(51.43) ) - 180 ?  Yes.

 

Your figure has 7 sides.  It is a heptagon.

Answer 2

If you walk around the perimeter of any polygon with one arm lifted to point in the direction of your walk, each time you step onto a new side and face in its direction your arm will sweep out the exterior angle of that vertex. When you get back to your starting point your arm will have swept out all the exterior angles, adding them, and be pointing in the same direction as when you started. You’ve just measured the sum of the exterior angles of the polygon as 360°.

Using the vertices of a polygon as vertices of triangles, n - 2 triangles can be drawn inside a polygon, where n is the number of sides of the polygon. The sum of the interior angles of each of those triangles is 180°, so the sum of the interior angles of the polygon is 180°(n - 2).

The problem says, “the sum of the measures of the interior angles of a regular polygon is 180 degrees less than three times the sum of the measures of its exterior angles. How many sides does the polygon have?”, so:

180°(n - 2) = 3 (360°) - 180°

180°(n - 2) = 3 (2 * 180°) - 180°

180°(n - 2) = 6 * 180° - 180°

180°(n - 2) = 180°(6 - 1)

n - 2 = 5

n = 7 sides, whether its regular or not.