Help with finding number of sides, if polygon exists

[Guest]

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.

Thank you.

 

[Denis]

Post the problem in FULL, please.

As example, a regular pentagon is made up of 5 identical isosceles triangles,
with one angle = 72 degrees and the other two = 54 degrees each; clear?

 

[soroban]

Hello, boblandersen!

Which 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°
22. 135°

How do I calculate the number of sides?

You are expected to know this formula . . .

If a polygon has \(\displaystyle n\) sides, the sum of its interior angles is: \(\displaystyle \,180(n\,-\,2)\) degrees.

If the polygon is regular (equal sides and equal angles),
\(\displaystyle \;\;\) then each angle has: \(\displaystyle \,180\left(\frac{n\,-\,2}{n}\right)\) degrees.


\(\displaystyle 21)\;150^o\)

We have: \(\displaystyle \,180\left(\frac{n\,-\,2}{2}\right) \;= \;150\)

Multiply by \(\displaystyle n:\;\;180(n\,-\,2)\; =\;150n\)

We have:\(\displaystyle \,180n\,-\,360\;=\;150n\;\;\Rightarrow\;\;30n\:=\:360\)

Therefore: \(\displaystyle \,n\:=\:12\) . . . a regular dodecagon has an interior angle of 150°.


\(\displaystyle 22.\;135^o\)

We have: \(\displaystyle \,180\left(\frac{n\,-\,2}{n}\right)\;=\;135\)

Multiply by \(\displaystyle n:\;\;180(n\,-\,2)\;=\;135n\)

We have: \(\displaystyle \,180n\,-\,360\;=\;135n\;\;\Rightarrow\;\;45n\:=\:360\)

Therefore: \(\displaystyle \,n\:=\:8\) . . . a regular octagon as an interior angle of 135°.


Everything is easier if you have the right tools . . .

 

[skeeter]

Everything is easier if you have the right tools . . .

indeed, it is ...

your initial approach using the formula a = 180 - 360/n was excellent thinking!

the exterior angles of any regular polygon ...
1. are congruent
2. sum to 360 degrees
3. are supplementary to their respective interior angles

if an interior angle is 150 degrees, then the exterior angle is 30 degrees ...
360/30 = 12 exterior angles -> a dodecagon.

if an interior angle is 135 degrees, then the exterior angle is 45 degrees ...
360/45 = 8 exterior angles -> an octagon

but, let's look at the original question ... are there some measures for interior angles that cannot possibly form a regular polygon?
how about 130 degrees? ... the exterior angle would be 50 degrees, but 360/50 does not work out to be whole number. So, no regular polygon exists whose interior angle is 130 degrees.

 

[TchrWill]

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.