I have to answer the following question on a test and need step by step help, I don't understand and can't figure out how to answer it.
Question: Wach quantity might be the measure of one interior angle of a regular polygon. Find, if possible the number of sides of the polygon. If no such polygon exists, so indicate.
21. 150 degrees
22. 135 degrees
How do I calculate the number of sides?
I think using "a = 180 degrees - 360 degrees/n". If that is the formula that I should use, please show me how, step by step.
Polygons
A polygon is a plane figure with three or more line segments and angles that are joined end to end so as to completely enclose an area without any of the line segments intersecting.
A convex polygon is one where the line segments joining any two points of the polygon remain totally inside the polygon, each interior angle being less than 180º.
A concave polygon is one where one or more line segments joining any two points of the polygon are outside of the polygon and one or more of the interior angles is greater than 180º. The inward pointing angle of a concave polygon is referred to as a reentrant angle. The angles less than 180º are called salient angles.
A regular polygon is one where all the sides have the same length and all the interior angles are equal.
A diagonal is a straight line connecting any two opposite vertices of the polygon.
Polygons are classified by the number of sides they have.
No. of sides.........Polygon Name
......3.....................Triangle
......4..................Quadrilateral
......5....................Pentogon
......6....................Hexagon
......7....................Heptagon
.....8......................Octagon
.....9......................Nonagon
....10.....................Decagon
....11....................Undecagon
....12....................Dodecagon
....13....................Tridecagon
....14....................Tetradecagon
....15....................Pentadecagon
......n........................n-gon
Regular Polygon Terminology
n = the number of sides
v = angle subtended at the center by one side = 360/n
s = the length of one side = R[2sin(v/2)] = r[2tan(v/2)]
R = the radius of the circumscribed circle = s[csc(v/2]/2 = r[sec(v/2)]
r = the radius of the inscribed circle = R[cos(v/2)] = s[cot(v/2)]/2
a = apothem = the perpendicular distance from the center to a side (the radius of the inscribed circle)
p = the perimeter = ns
Area = s^2[ncot(v/2)]/4 = R^2[nsin(v)]/2 = r^2[ntan(v/2)]
The formula for the area of a regular polygon is also A = (1/2 )ap = (1/2)ans, where a is the apothem, p is the perimeter, s is the side length and n is the number of sides..
The sum of all the interior angles in a polygon is 180(n - 2)
The sum of the exterior angles in a polygon is 360º.
The internal angle between two adjacent sides of a regular polygon is given by 180(n - 2)/n
The external angle between any side and the extended adjacent side of a regular polygon is given by 360/n.
You might be interested in why the sum of all the interior angles of a polygon is 180(n - 2).
Consider first the square, rectangle and trapazoid. Draw one ofthe diagonals in each of these figures.
What is created is two triangles within each figure.
The sum of the interior angles of any triangle is 180 deg.
Therefore, the sum of the interior angles of each of these 4 sided figues is 360 Deg.
Now consider a pentagon with 5 sides that can be divided up into 3 triangles.
Therefore, the sum of the interior angles of a pentagon is 540 Deg.
What about a hexagon. I tink you will soonsee that the sum of the interior angles is 720 Deg.
Do you notice anything?
n = number of sides........3........4........5........6
Sum of Int. Angles.........180....360....540....720
The sum of the interior angles is representable by 180(n - 2).
Consider also the sum of the exterior angles.
Each exterior angle is 180 - 180(n - 2)/n = (180 - 180n + 360)/n = 360/n.
Therefore, the sum of the exterior angles is 360n/n or 360 Deg.
Thank you.
