SOLUTION: find the measure of a vertex angle, a central angle, and an exterior angle for a regular 40-gon figure Asked other tutors and my teacher. still confusing me. any help appreciate

Question 1139141: find the measure of a vertex angle, a central angle, and an exterior angle for a regular 40-gon figure
Asked other tutors and my teacher. still confusing me. any help appreciated. Thanks!
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
a 40-gon figure is a figure that has 40 sides.

by the way, a 40-gon figure is called a Tetracontagon.

here's a link that has all the names of the gons, or at least a whole lot of them.

https://en.wikipedia.org/wiki/List_of_polygons

you have to scroll about halfway down the page to get to the gon names.

the sum of the interior angles of a polygon is equal to (n-2) * 180.

for example, the sum of the interior angles of a triangle is (3-2) * 180 = 180 and the sum of the interior angles of a quadrilateral is (4 - 2) * 180 = 360 and the sum of the interior angles of a pentagon is (5 - 2) * 180 = 540.

when we get to the tetracontagon, the sum of the interior angles would be (40 - 2) * 180 = 6840 degrees.

each interior angle of a regular tetracontagon would be 6840 / 40 = 171 degrees.

you can equate this procedure to finding the value of each interior angle of a regular quadrilateral = 360 / 4 = 90 degrees, or a regular triangle = 180 / 3 = 60 degrees.

a regular polygon is defined as a polygon that has all internal angles equal and all sides equal.

enough with internal angles.

each internal angles of a tetracontagon is equal to 171 degrees.

that would make each external angle of a tetracontagon equal to 180 - 171 = 9 degrees.

the other way to find the value of each external angle of a polygon is to divide 360 by the number of sides.

for a triangle, each external angle would be 360 / 3 = 120 degrees.
this stands to reason because each internal angle is 60 degrees and each external angle is equal to 180 minus the value of each internal angle = 180 - 60 = 120.

with the tetracontagon, each external angle is therefore equal to 360 / 40 = 9 degrees.

i believe the central angle would be the same as the external angle.

that would make it equal to 9 degrees.

so, the general formulas for a regular polygon are:

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

each interior angle is (n-2) * 180 / n, where n is the number of sides.

each exterior angle is 360 / n, where n is the number of sides.

each central angle is 360 / n, where n is the number of sides (same formula as for each exterior angle).

the interior angle is also called the vertex angle.

there are numerous references to tetracontagons on the web.

here's a sample of a search page i did on google.

https://www.google.com/search?q=tetracontagon&rlz=1C1CHBF_enUS839US839&oq=tetracontagon&aqs=chrome.0.69i59l2j69i60l3j0.1823j0j4&sourceid=chrome&ie=UTF-8

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