Polygon angles (basic)

[McEwan25]

Hi there,

I am answering a question in my workbook and not sure where I am going wrong. The question is:

'The size of each interior angle of a regular polygon is 168°. How many sides has the polygon?'

Now how I went about answering this is using the angle facts explained earlier in the chapter.


If the interior angles are all 168° and I have to work out how many sides it has, what I did was to:

1) Multiply 168 in sets of 5 to find an answer that would divide evenly by 180° (because the number of triangles in the polygon will be number of sides -2). Therefore I can just find that 168x15 = 2520. 2520 ÷ 180 = 14 triangles, or 16 sides.

Therefore a regular polygon with interior angles of 168° will have 16 sides. But the answer page says 30. Where am I going wrong here?

I did the same process with the previous question which used 140° interior angles and got the correct answer, 9.

Any help much appreciated, thanks.

 

[Deleted member 4993]

Hi there,

I am answering a question in my workbook and not sure where I am going wrong. The question is:

'The size of each interior angle of a regular polygon is 168°. How many sides has the polygon?'

Now how I went about answering this is using the angle facts explained earlier in the chapter.


If the interior angles are all 168° and I have to work out how many sides it has, what I did was to:

1) Multiply 168 in sets of 5 to find an answer that would divide evenly by 180° (because the number of triangles in the polygon will be number of sides -2). Therefore I can just find that 168x15 = 2520. 2520 ÷ 180 = 14 triangles, or 16 sides.

Therefore a regular polygon with interior angles of 168° will have 16 sides. But the answer page says 30. Where am I going wrong here?

I did the same process with the previous question which used 140° interior angles and got the correct answer, 9.

Any help much appreciated, thanks.

{ am not sure exactly what you are doing - but I do it this way:

If the polygon is divided into several (n) isosceles triangles - the central angle of each triangle is 180 - 168 = 12

No of triangles = 360/12 = 30

So we have a 30 sided regular-polygon.

 

[McEwan25]

Hmm. Not quite sure I get you, but I managed to work out that 180-168 = 12° exterior angle. And that all exterior angles must = 360 therefore the number of sides will be 360÷12= 30.

Am I safe to use this second method in future?

 

[Deleted member 4993]

Hmm. Not quite sure I get you, but I managed to work out that 180-168 = 12° exterior angle. And that all exterior angles must = 360 therefore the number of sides will be 360÷12= 30.

Am I safe to use this second method in future?

You were given the interior angle of the regular polygon.

Look at:

How to Find Degrees in Polygons

A polygon is a closed two-dimensional shape composed of three or more connected line segments. Triangles, trapezoids and octagons are common examples of polygons. Polygons are typically classified according to number of sides and the relative measures of its sides and angles. They also are...

sciencing.com

Don't cram it - try to understand the derivation. Why (180 - 168) and why (360 ÷ 12)?

 

[McEwan25]

Perhaps my textbook isn't giving me every way to work this out. I can't understand why:

'the central angle of each triangle is 180 - 168 = 12

No of triangles = 360/12 = 30

So we have a 30 sided regular-polygon.'

For what it is worth I've only started studying polygons and basic geometry a couple of hours ago...

Are you saying that the 'central angle' is 12 because the three angles of an isosceles triangle = 180 and that 168 must equal the sum of the other two angles? Therefore 84+84+12 = 1 isosceles triangle?

If so I understand that in a vacuum, but don't understand the translation to this:

Where are we getting the number of triangles = 360/12 = 30. I thought we can only work out the number of triangles once we know the number of sides? (N-2)


I did the second method because I know that the sum of external angles must equal 360° and the internal and external angles are supplementary, therefore all exterior angles must equal 180° minus the interior angle... so 180-168 = 12°.

And if there are x number of 12° exterior angles in 360° all I need to do is find out x which is easy because there are as many exterior angles as there are sides. So 360/12 = 30 sides.

I am struggling to visualise your method. Apologies.

 

[pka]

If the interior angles are all 168° and I have to work out how many sides it has, what I did was to:
1) Multiply 168 in sets of 5 to find an answer that would divide evenly by 180° (because the number of triangles in the polygon will be number of sides -2). Therefore I can just find that 168x15 = 2520. 2520 ÷ 180 = 14 triangles, or 16 sides.
Therefore a regular polygon with interior angles of 168° will have 16 sides. But the answer page says 30. Where am I going wrong here?

Here is a link that gives the relation between the edges of a regular polygon and the measure each interior angle.
Applying the formula: \(\dfrac{180(n-2)}{n}=168\\180n-360=168n\\12n=360\\n=30\)

 

[McEwan25]

OK thank you, I'll give that a good read over.

 

[Steven G]

1) Multiply 168 in sets of 5 to find an answer that would divide evenly by 180° (because the number of triangles in the polygon will be number of sides -2). Therefore I can just find that 168x15 = 2520. 2520 ÷ 180 = 14 triangles, or 16 sides.

I have no idea where the 5 comes from. I will tell you that there are many many multiples of 168 that can be divided evenly by 180. How about 2250, 4500, 6750, 9000, ...
If your method is to work you would need to know which one to pick.

 

[McEwan25]

I have no idea where the 5 comes from. I will tell you that there are many many multiples of 168 that can be divided evenly by 180. How about 2250, 4500, 6750, 9000, ...
If your method is to work you would need to know which one to pick.

I was thinking that the 168 would have to be multiplied in 5s so that the units column (8) will be a number that is divided by 180 evenly.

So for example 168x7 is 1,176 which obviously can't be divided by 180 evenly because it needs to end in a 0, as no multiples of 180 will have a 1,2,3,4,5,6,7,8 or 9 in the units column. Therefore it must go in multiples of 5.


In saying that I now realise I was making things far too complex for myself.... I'll use the second method

 

[Steven G]

I was thinking that the 168 would have to be multiplied in 5s so that the units column (8) will be a number that is divided by 180 evenly.

So for example 168x7 is 1,176 which obviously can't be divided by 180 evenly because it needs to end in a 0, as no multiples of 180 will have a 1,2,3,4,5,6,7,8 or 9 in the units column. Therefore it must go in multiples of 5.


In saying that I now realise I was making things far too complex for myself.... I'll use the second method

Only half of the multiples of 5 ends in a 0. The other half ends in 5. (If your method works-which it doesn't) why not use 10 instead of 5? After all every multiply of 10 ends in 0 and every number that ends in 0 is a multiple of 10.

 

[McEwan25]

Only half of the multiples of 5 ends in a 0. The other half ends in 5. (If your method works-which it doesn't) why not use 10 instead of 5? After all every multiply of 10 ends in 0 and every number that ends in 0 is a multiple of 10.

Because it isn't multiples of 5, it is multiples of 8, the only ones of which end in a 0 are 5, 10, 15, 20 and so on.

Thank you for the continued discussion. I realise that I was making it far too complex.