How many sides.

Question

The sum of interior angles is 110.  Using the formula 180(n-2)/n How many sides does it have. 

Answer 1

In a three sided figure, n = 3, the sum of the angles is 180.

Since 110 < 180 such a figure does not exist.

 

Answer 2

In the 1960's Seymour Papert with a team at MIT invented "Turtle Graphics" (Logo computer language) to teach children about computers.  The turtle on the floor responded to commands to turn, move, pen up/down, etc.

 

Now, to create an n-sided figure, the Turtle moves, turns the first exterior angle, moves, turns the second exterior angle, ... and gets to the starting point and finally turns back to the starting direction.  The turtle's turns always total 360 degrees (it turned completely around).

 

The interior angles, being the supplement of the exterior angles, are (180-x) degrees.  So, adding them up, depends on the number of turns (also sides) made in going around the perimeter of the shape.

 

An easy example -- a square

   Move 1, Turn Left 90, move 1, turn left 90, move 1, turn left 90, move 1, turn left 90

    The total of the exterior angles is 360

    The total of the interior angles is (180-90)+(180-90)+(180-90)+(180-90)

    A square has 4 sides, so (180)(n-2)/n = (180)(2)/4  = 90  (each interior angle is 90 degrees)

 

In this problem, (180)(n-2)/n = interior angle

            and     (180)(n-2)  = n*angle = sum of the angles = 110

 

  (180)(n-2) = 110

 Now, even with n=3, as Michael J. pointed out, there wont be a closed figure.  That 110 value must be a multiple of 180 and it must be at least 180 (that is a triangle).

 

FYI:  If you want to draw some figures using Logo, there is an online web site at:

      http://turtleacademy.com/

 

Answer 3

The sum of the interior angles is represented by

 

180(n - 2)

 

where n is the number of sides.

 

 

180(n - 2) = 110

 

Solve for n from this equation.

 

 

Fun fact:

 

A closed figure has interior angles that add up to 180 and greater.  Therefore, any interior angles that add up to less than 180 does not constitute a closed figure.