Suppose the polygon has V vertices.Then sum of interior angles is (V - 2)*180 degrees = 1980 degrees
=> V - 2 = 1980/180 = 11 => V = 13
A polygon with V vertices has V*(V-3)/2 = 13*10/2 = 65 diagonals.
If it is a regular polygon with each interior angle of 120 degrees then it is an hexagon which will have 0.5((6^2)-3*6) equals 9 diagonals
nonagon
It is a 6 sided hexagon polygon Its sides are of different lengths Its interior angles are of different sizes Its 6 interior angles add up to 720 degrees Its 6 exterior angles add up to 360 degrees It has 9 diagonals It can't tessellate
Well, honey, a polygon with an interior angle of 10 degrees would have 36 sides. Each interior angle of a polygon can be calculated using the formula (n-2) * 180 / n, where n is the number of sides. So, in this case, you would solve for n by setting 10 = (n-2) * 180 / n. Math can be a real hoot sometimes!
If the polygon has n sides, the sum of its interior angles is 180*(n - 2) degrees. [Whether the polygon is regular or not is irrelevant] So, 180*(n - 2) = 1080 (n - 2) = 6 n = 8 Then number of diagonals = n*(n - 3)/2 = 8*5/2 = 20
If an interior angle is 40 degrees, the polygon must be irregular since the smallest angle for a regular polygon is 60 degrees. Since it is irregular, the polygon can have any number of sides.
If it is a regular polygon with each interior angle of 120 degrees then it is an hexagon which will have 0.5((6^2)-3*6) equals 9 diagonals
It cannot be a regular polygon with interior angles of 24 degrees since the smallest interior angle in a regular polygon is 60 degrees. If it is an irregular polygon, then any one of them, with 3 or more sides, can have an angle of 24 degrees.
The sum of the interior angles of any convex polygon with n sides is n * 60 degrees. Any given interior angle, however, can be anything from just greater than 0 degrees to just less than 180 degrees.
A convex polygon.
nonagon
It is a 6 sided hexagon polygon Its sides are of different lengths Its interior angles are of different sizes Its 6 interior angles add up to 720 degrees Its 6 exterior angles add up to 360 degrees It has 9 diagonals It can't tessellate
Well, honey, a polygon with an interior angle of 10 degrees would have 36 sides. Each interior angle of a polygon can be calculated using the formula (n-2) * 180 / n, where n is the number of sides. So, in this case, you would solve for n by setting 10 = (n-2) * 180 / n. Math can be a real hoot sometimes!
An irregular pentagon.
If the polygon has n sides, the sum of its interior angles is 180*(n - 2) degrees. [Whether the polygon is regular or not is irrelevant] So, 180*(n - 2) = 1080 (n - 2) = 6 n = 8 Then number of diagonals = n*(n - 3)/2 = 8*5/2 = 20
That would depend if it's a regular or an irregular polygon but in both cases all the interior angles add up to 3420 degrees
Any irregular polygon does not have equal sides and equal interior angles