# Polygons at a Point

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• May 27th 2006, 10:53 PM
confused
a scalene and isosceles triangle are also "regular" - don't tell me they aren't cause last yr in maths comp, a scalene was considered "regular" and i put it down as irregular and i got it wrong.
• May 27th 2006, 11:00 PM
CaptainBlack
Quote:

Originally Posted by confused
a scalene and isosceles triangle are also "regular" - don't tell me they aren't cause last yr in maths comp, a scalene was considered "regular" and i put it down as irregular and i got it wrong.

It is not only me telling you that a regular 3-gon is an equilateral triangle,
see MathWorld.

Also, your pentagon is not regular.

RonL
• May 27th 2006, 11:03 PM
confused
then i can only conclude that the board of studies are bull****
• May 27th 2006, 11:04 PM
CaptainBlack
Quote:

Originally Posted by confused
then i can only conclude that the board of studies are bull****

It is not unknown.

RonL
• May 27th 2006, 11:05 PM
confused
i can't even think why skool is "compulsory".....so sad
• May 28th 2006, 09:25 AM
Soroban
Hello, Chuck_3000!

Quote:

Certain sets of regular polygons fill space around a point without gaps or overlapping.
e.g. imagine 3 hexagons joined together and the common vertice is the point they surround.
That is classified a [6,6,6] point fill set (3 six sided shapes)

Although the polygons can be arranged in a different order, we count these as the same.

We call a set of regular polygons filling space around a point a point-fill set.

a) Find a point-fill set of 3 polygons containing a 24-gon
b) Explain why it is not possible to have a point fill set containing a triangle and a pentagon
c) Find all point-fill sets that contain at least one square.

The interior angle of a regular $\displaystyle n$-gon is: .$\displaystyle \frac{n-2}{n}\cdot180^o$

Armed with this formula, you can make a list:

$\displaystyle \text{Triangle: }60^o\quad\text{Square: }90^o\quad\text{Pentagon: }108^o\quad\text{Hexagon: }120^o$

$\displaystyle \text{Octagon: }135^o\quad\text{Decagon: }144^o\quad\text{12-gon: }150^o\quad \text{15-gon: }156^o$

$\displaystyle \text{18-gon: }160^o\quad\text{20-gon: }162^o\quad\text{24-gon: }165^o$

$\displaystyle \text{30-gon: }168^o\quad\text{36-gon: }170^o$

(a) A 24-gon takes up $\displaystyle 165^o$ of the circle about the point,
. . leaving $\displaystyle 360^o - 165^o \:=\:195^o$ to be filled.
This can be accomplished with a triangle $\displaystyle (60^o)$ and an octagon $\displaystyle (135^o)$.

(c) Of course, four squares comprise a point-fill set.

With two squares, there are $\displaystyle 180^o$ to be filled.
This can be accomplished with:
. . three triangles: $\displaystyle 3 \times 60^o$
. . a hexagon and a triangle: $\displaystyle 120^o + 60^o$

With one square, there are $\displaystyle 270^o$ to be filled.
This can be accomplished with:
. . two triangles and a 12-gon: $\displaystyle 60^o + 60^o + 150^o$
. . a hexagon and a 12-gon: $\displaystyle 120^o + 150^o$
. . two octagons: $\displaystyle 135^o + 135^o$
. . a pentagon and a 20-gon: $\displaystyle 108^o + 162^o$

I hope I didn't miss any . . .
• May 30th 2006, 09:09 AM
MathGuru
The question asked in an Australian take-home competition. In case you wanted just to check, it is the:
2006 Maths Challenge Stage
Mathematics Challenge for Young Australians
TERM 2

JUNIOR STUDENT PROBLEMS

An Activity of the Australian Mathematical Olympiad Committee
A Subcommittee of the Australian Mathematics Trust in association with the Australian Academy of Science and the University of Canberra

...Just google Australian Mathematics Trust Maths Challenge or something like that ;)

This person happened to be trying to complete one of the hardest questions in the book - well, it's hard until you figure out the key point to it, then it's easy. I'm just not satisfied this thread is still up there, and she/he's been given the answers. Unfortunately/Fortunately, I don't think she/he'll be able to write a 1 to 2 page explanation on Egyptian algebra and other techniques =) Especially not since they're probably about 13. There's another easier way...

This competition is due on Wednesday ((GMT+10:00 Sydney))...so it won't make much of a difference if it isn't deleted or not, just as long as no-one else posts answers anymore.
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