request for info

I'd be grateful for some basic help in the geometry department. I'm planning
to map out some lattices for an event. What's the inside angle of a 10 sided
polygon and how do you arrive at the answer?

thanks

mopani (not the sharpest pin in the box)

It's easy to see (or grasp the notion that) the interior angles of a
triangle sum up to 180º

Any quadrilateral can be divided (partitioned) into two triangles.
Any pentagon can be divided into three triangles. And so on...

Thus, the sum of interior angles of a n-sided polygon [n-gon] is (n-
2)x180º

If we assume we're speaking of regular polygons, we then have that
an interior angle of an n-gon has

[(n-2)x180º]/n degrees, or 180º - 360º/n degrees.

So, for a (regular) decagon, we conclude that the interior angle has
144 degrees.

Max.

--- In tuning@yahoogroups.com, <mopani@t...> wrote:
> I'd be grateful for some basic help in the geometry department.
I'm planning
> to map out some lattices for an event. What's the inside angle of
a 10 sided
> polygon and how do you arrive at the answer?
>
> thanks
>
> mopani (not the sharpest pin in the box)

--- In tuning@yahoogroups.com, <mopani@t...> wrote:

> I'd be grateful for some basic help in the geometry department. I'm
planning
> to map out some lattices for an event. What's the inside angle of a
10 sided
> polygon and how do you arrive at the answer?

The vertex angle of a regular n-gon is (n-2)/n times pi radians, or
equivalently (n-2)/n 180 degrees; for a regular 10-gon (decagon) that
is (8/10) 180 = 144 degrees.

on 14/3/05 16:49, Maximiliano G. Miranda Zanetti at
giordanobruno76@yahoo.com.ar wrote:

>
>
> It's easy to see (or grasp the notion that) the interior angles of a
> triangle sum up to 180º
>
> Any quadrilateral can be divided (partitioned) into two triangles.
> Any pentagon can be divided into three triangles. And so on...
>
> Thus, the sum of interior angles of a n-sided polygon [n-gon] is (n-
> 2)x180º
>
> If we assume we're speaking of regular polygons, we then have that
> an interior angle of an n-gon has
>
> [(n-2)x180º]/n degrees, or 180º - 360º/n degrees.
>
> So, for a (regular) decagon, we conclude that the interior angle has
> 144 degrees.
>
> Max.
>

Thanks Max, you're a star.

mopani

Mopani,

I take it you mean a _regular_ polygon, one with
equal sides and angles?

If so, you can work it out by observing this pattern -

Sides Polygon Sum Interior angle
------ --------- ----- ----------------
3 triangle pi pi/3
4 square 2 pi 2 pi/4
5 pentagon 3 pi 3 pi/5
6 hexagon 4 pi 4 pi/6
...
10 decagon 8 pi 8 pi/10
...

- where pi radians is 180 degrees.

In the case of your decagon, you have an interior
angle of 144 degrees.

Regards,
Yahya

-----Original Message-----
________________________________________________________________________
Date: Mon, 14 Mar 2005 12:13:31 +0100
Subject: request for info

I'd be grateful for some basic help in the geometry department. I'm planning
to map out some lattices for an event. What's the inside angle of a 10 sided
polygon and how do you arrive at the answer?

thanks

mopani (not the sharpest pin in the box)

________________________________________________________________________

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.308 / Virus Database: 266.7.2 - Release Date: 11/3/05

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:

> It's easy to see (or grasp the notion that) the interior angles of a
> triangle sum up to 180º

That requires another proof. I imagine the easiest way to get the
formula for the interior angle of an n-gon is that it is 180 - n/360
degrees; so 360/n is the amount by which one side turns to get to the
next, and n of the 360/n add up to a complete 360 degree turn.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:
>
> > It's easy to see (or grasp the notion that) the interior angles
of a
> > triangle sum up to 180º
>
> That requires another proof. I imagine the easiest way to get the
> formula for the interior angle of an n-gon is that it is 180 -
n/360
> degrees; so 360/n is the amount by which one side turns to get to
the
> next, and n of the 360/n add up to a complete 360 degree turn.

Humm... have you read my message? There I tried to give an intuitive
approach to the formula.

In regard to the "postulatum" of 180º for a triangulum, you are
completely right that it's demonstrable. Maybe it was just too much
geometry for a day :D

Max.