Fun problem! (1 Viewer)

[Carrotsticks]

Consider a unit square.

From one corner to the other, a circular arc is drawn.

This is repeated 4 times, and a small 4 sided figure is created from the intersections of these arcs.

Find the area of this 4 sided figure (it should look like a square with curved edges).

EXTENSION:

The same question as above, but with a Regular Pentagon.

 

[math man]

for square (i) for n gon (ii)



am i close?

 

[Carrotsticks]

 

[seanieg89]

I got:



for the square.

 

[seanieg89]

For a regular n-gon I am unsure of your definition of this region, as we we can draw several circular arcs containing two vertices centred at any given vertex...

Do you simply mean the region that consists of all points with distance at most one from every vertex?

 

[Carrotsticks]

I got:



for the square.

Yep, I got the same answer too.

I actually made a mini-document showing a way of doing it using only methods taught in Year 10 such as area of triangle, circle, soh cah toa etc (uncanningly, the file was made exactly 1 year ago). I managed to dig it up from deep within my hard drive. The days before I knew LaTeX...

View attachment Question.pdf

For a regular n-gon I am unsure of your definition of this region, as we we can draw several circular arcs containing two vertices centred at any given vertex...

Do you simply mean the region that consists of all points with distance at most one from every vertex?

Yep exactly. So here is the case for the Pentagon:

I haven't actually dug into the question yet, but methinks that there exists some N such that the area fails to exist because if we have a very large polygon with side length 1, most surely such a thing cannot exist.

So I guess the extension could be for the Pentagon.

 

[seanieg89]

I get the feeling it should always exist, the area will just get smaller and smaller. Have something I've got to do right now but will look at this again later. I'm not sure if my method is convenient to adapt to n-gons.

Okay, I have worked out a different method that should work for n-gons. Will post it here tonight if it is indeed correct.

EDIT: You are right about the limitations, n=<5.

 

[mirakon]

Yes I got da first part!

N-sided polygon totally stumps me though, can't even fathom how it works lol

 

[barbernator]

EDIT: You are right about the limitations, n=<5.

also, when n=6, all circles intersect at 1 point

 

[seanieg89]

Mine came out numerically as approximately 0.0790, although I am not entirely confident in it.

 

[barbernator]

Mine came out numerically as approximately 0.0790, although I am not entirely confident in it.

I had that originally!! but then i changed my solution cos I had something wrong with it and then i got my other solution, but maybe I was right the first time...

 

[seanieg89]

What was wrong?

 

[barbernator]

What was wrong?

shit, i dont even remember, ill have a look through my solution sheets.

 

[seanieg89]

Cool, will come back later and look through my working more carefully.

 

[barbernator]

shit, i dont even remember, ill have a look through my solution sheets.

I re-did the question, and i did get 0.0790 again. Dunno why i changed it originally