stumped on a octagon

[myshop1044]

attached are 2 ptn's, 1 has an octagon with all sides equal and angles are equal, clock wheel 2 ptn, I have the 4 sides top, bot, left and right equal in length, but the 4 angle sides are not the same length. the angles all equal 135 deg. I have the "grid snap on" and all are centered both ways, I can not figure how to get the all sides equal. Help me out on some old geometry if you would please.
Feel free to adjust the clock wheel 2, but let me know what you did ok.
been out of school to long.

Myshop

[AskBud]

Your attachments are missing.
AskBud

[cnsranch]

Let me see if I can clear up this little math problem.....

If you draw a circle of radius r and construct the octagon to lie
inside the circle with the vertices on the circumference of the
circle, then drop a perpendicular from the centre of the circle to the
mid-point of one of the sides, it is easy to see that the length of
this perpendicular is

r.cos(22.5), so width of octagon is 2r.cos(22.5)

Also, half the length of the side of the octagon is r.sin(22.5), so
the length of a side of the octagon is 2r.sin(22.5)

ratio side/width = 2rsin(22.5)/2rcos(22.5)

= tan(22.5) = 0.414213

You could stop here if you like, but if you would prefer the answer in
surd form, we can use the double angle formula for tan(x)

tan(45) = 2tan(22.5)/[1-tan^2(22.5)]

If tan(22.5) = x we get

1 = 2x/(1-x^2)
so 1-x^2 = 2x
x^2 + 2x - 1 = 0
x = [-2 +or- sqrt(4+4)]/2
x = [-2 +or- sqrt(8)]/2

we can ignore the negative option since x must be positive, so

x = [-2 + 2sqrt(2)]/2
= -1 + sqrt(2)

So ratio side/width = sqrt(2)-1 = 0.414213 as before.

Knowing the ratio, then if given the width the side =(sqrt(2)-1)width.
Given the side, then width = side/[sqrt(2)-1]

And if you think I came up with that, I've got some really great oceanfront property here in KC to have you take a look at.

[atauer]

Oh Lord!!!!

Jerry, is there gonna be a test on this?

[cnsranch]

Sorry, dude, couldn't resist.

My engineer son just walked in - here's his simple answer.....

Start with a circle, not a square. Divide the circle into eight "pie slices", the angle of each side of each slice is 22.5 degrees. Start with the top two sides, and go 22.5 degrees from vertical - the others will be 22.5 +22.5, etc.

Start at the center of the circle, and take the sides of the slices to the edge of the circle. Note the lengths, and constrain them to be equal in length.

Then using the connected lines tool, draw your octagon, intersecting the pie slices.

Either that, or get an engineer son (smile)

[AskBud]

41679
attached are 2 ptn's, 1 has an octagon with all sides equal and angles are equal, clock wheel 2 ptn, I have the 4 sides top, bot, left and right equal in length, but the 4 angle sides are not the same length. the angles all equal 135 deg. I have the "grid snap on" and all are centered both ways, I can not figure how to get the all sides equal. Help me out on some old geometry if you would please.
Feel free to adjust the clock wheel 2, but let me know what you did ok.
been out of school to long.

Myshop
Your Deer Clock had several angles not set to 135. I had to remove the lock on the ones that were 135, and then set/lock on of the mavericks to 135, unlock it and then proceed to the next maverick.

The second MPC has unequal sided. I'll work on it next.
AskBud

[atauer]

Sorry, dude, couldn't resist.

My engineer son just walked in - here's his simple answer.....

Start with a circle, not a square. Divide the circle into eight "pie slices", the angle of each side of each slice is 22.5 degrees. Start with the top two sides, and go 22.5 degrees from vertical - the others will be 22.5 +22.5, etc.

Start at the center of the circle, and take the sides of the slices to the edge of the circle. Note the lengths, and constrain them to be equal in length.

Then using the connected lines tool, draw your octagon, intersecting the pie slices.

Either that, or get an engineer son (smile)

Just as a FYI, I actually have a degree in computer/electrical engineering.

I just hate long drawn out equations. I can do them, but hate to do them...

[PCW]

Let me see if I can clear up this little math problem.....

If you draw a circle of radius r and construct the octagon to lie
inside the circle with the vertices on the circumference of the
circle, then drop a perpendicular from the centre of the circle to the
mid-point of one of the sides, it is easy to see that the length of
this perpendicular is

r.cos(22.5), so width of octagon is 2r.cos(22.5)

Also, half the length of the side of the octagon is r.sin(22.5), so
the length of a side of the octagon is 2r.sin(22.5)

ratio side/width = 2rsin(22.5)/2rcos(22.5)

= tan(22.5) = 0.414213

You could stop here if you like, but if you would prefer the answer in
surd form, we can use the double angle formula for tan(x)

tan(45) = 2tan(22.5)/[1-tan^2(22.5)]

If tan(22.5) = x we get

1 = 2x/(1-x^2)
so 1-x^2 = 2x
x^2 + 2x - 1 = 0
x = [-2 +or- sqrt(4+4)]/2
x = [-2 +or- sqrt(8)]/2

we can ignore the negative option since x must be positive, so

x = [-2 + 2sqrt(2)]/2
= -1 + sqrt(2)

So ratio side/width = sqrt(2)-1 = 0.414213 as before.

Knowing the ratio, then if given the width the side =(sqrt(2)-1)width.
Given the side, then width = side/[sqrt(2)-1]

And if you think I came up with that, I've got some really great oceanfront property here in KC to have you take a look at.

Any one catch what he said??? <smile>

[AskBud]

41679
Your Deer Clock had several angles not set to 135. I had to remove the lock on the ones that were 135, and then set/lock on of the mavericks to 135, unlock it and then proceed to the next maverick.

The second MPC has unequal sided. I'll work on it next.
AskBud

The 2nd MPC had 4 sides at 5.188 and 4 sides at 4.402.
I added 5.188 to 4.402 to get 9.590. I then divided that by 2 to get 4.795. I next set all sides to 4.795. Once that was done, I began setting the angles to 135 degrees.
AskBud
41681

[myshop1044]

Bud wins the prize, thanks you fellows, ya'll made my head hurt with all those answers. I like pratical math.
Just one more question for Bud, how did you unlock the angles on the deer clock. I notice that when I tried to move the octagon on the deer clock, it would not allow me to move it up or down in size. I was able to move mine up and down in size before I edited the lengths and angles. Is it the variable lengths and angles the allows it to be raise up and down in size or the locked or unlocking?
just a little more info for me.

Myshop

[AskBud]

Bud wins the prize, thanks you fellows, ya'll made my head hurt with all those answers. I like pratical math.
Just one more question for Bud, how did you unlock the angles on the deer clock. I notice that when I tried to move the octagon on the deer clock, it would not allow me to move it up or down in size. I was able to move mine up and down in size before I edited the lengths and angles. Is it the variable lengths and angles the allows it to be raise up and down in size or the locked or unlocking?
just a little more info for me.

Myshop
If you look at my offering of the MPC, you see the dimensions on the sides are Yellow/Locked. Right click on each dimension and select "remove length constraint". Now, you may use the Yellow corner markers to enlarge or compress the figure. The sides will all change, equally, in size as you control the Yellow corner marker. Do not use the RED or Yellow dots as that will skew your figure.

If you look closely, at the MPC, you will also see that somehow the top-most and the Right-most sides also have a lock/constraint of 0.0. This effectively centered the figure, however it does not show up as centered when you review the "Center Icon" in the Layout Tab. If you wanted to move the figure right, left, up, or down, you would need to unlock those constraints as well.
AskBud