untrigintagonal

Untrigintagonal refers to a polygon with 31 sides and 31 angles. The term is derived from Latin and Greek roots, with "un" meaning one, "triginta" meaning thirty, and "gonal" referring to angles or sides. A polygon with 30 sides is called a trigintagonal polygon, and adding "un" signifies one more side, thus 31.

In geometry, an untrigintagonal polygon can be regular or irregular. A regular untrigintagonal polygon has all sides of equal length and all interior angles equal. The sum of the interior angles of any untrigintagonal polygon, regardless of regularity, is calculated using the formula (n-2) * 180 degrees, where n is the number of sides. For an untrigintagonal polygon, this would be (31-2) * 180 = 29 * 180 = 5220 degrees.

The measure of each interior angle of a regular untrigintagonal polygon can be found by dividing the sum of the interior angles by the number of sides: 5220 / 31 degrees. This results in an angle of approximately 168.39 degrees. Similarly, the measure of each exterior angle of a regular untrigintagonal polygon is 360 / 31 degrees, approximately 11.61 degrees.

The construction of a regular untrigintagonal polygon is complex due to its number of sides. It is not constructible with a compass and straightedge, as 31 is a prime number that is not a Fermat prime.