The sum of the measures of the interior angles of a polygon is 720 ∘ . What type of polygon is it?

Question

The sum of the measures of the interior angles of a polygon is       720          ∘       . What type of polygon is it?

lelestalis80d · Accepted Answer

Step 1Let&#039;s do this two ways: Using a pattern we know and using exterior angles.PatternFor a triangle, the sum of interior angles is 180 degrees.For a quadrilateral, the sum of interior angles is 360 degrees.We know that there is some pattern, so we can deduce quite logically that a pentagon has a sum of 540 degrees and a hexagon has a sum of 720 degrees, hence the answer is a hexagon.Exterior AnglesThe real question is how do we find that there is a pattern? We can recall that the sum of all exterior angles in any polygon is 360 degrees. If you don&#039;t remember learning this, you can imagine driving along the perimeter of any polygon and seeing that, in the end, you&#039;ve taken a single turn.Anyway, in a normal polygon with n sides, each of these exterior angles is, therefore,                     360        ∘            n       . Since an external angle plus an interior angle is a line, we know that an interior angle will have measure             180      ∘        -                  360        ∘            n      With n of those angles, we get the total internal angle of                     (        180        n        -        360        )            ∘       . Setting this equal to             720      ∘       , we easily find       n    =    6   .

Taylor Barron · Accepted Answer

Step 1A polygon of n number of sides can be divided into       n    -    2   number of triangles.Let the polygon have n number of sides then the sum of its interior angles will be equal to the sum of interior angles of       n    -    2   triangles      =          (      n      -      2      )              180      ∘                  720      ∘        =          (      n      -      2      )              180      ∘            n    -    2    =          720      180            n    -    2    =    4        n    =    6  Hence the polygon has 6 sides hence it&#039;s a hexagon

The sum of the measures of the interior angles of a polygon is 720^circ. What type of polygon is it?

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