Raised Bed Geometry Problem

I have a yard area that I want to put a circular raised be in. The area is sort of a wedge with the tip chopped-off, except that one of the sides (the west) is along side a driveway that flares into the area as it approaches the south side (the street).
The south side is 10' long. The east side is 31' long. The north side is 30' long. The west side is two segments, with a very obtuse angle (not quite a straight line); the north segment on the west side is 14' long, and the south segment is 10' long. The northwest and southeast corners are 90-degrees. The northeast corner is a little acute. The southwest corner is a little obtuse.
I want to create the biggest circle I can that touches the east, south (the street), and west (the driveway) sides. (It should not touch the north side.)
If things don't add-up right, keep in mind that the measurements were rounded to the nearest half-foot, and the exact location of the northeast corner could be off slightly.
How big should the circle be?
My guess is that it's going to be around 11' or 12' in diameter. The problem is that because of what's planted where, and when I can do the work, I'll need to dig-up, and layout the northeast 2/3 of the circle before I can clear the southwest 1/3. I can guess at a center point, and sweep a string along that 2/3 of the northeast side, but I can't sweep it over the southwest 1/3 to be sure it touches both the south and the west side, but doesn't go over either. There's only so much I'll be able to fudge the circle without people noticing it's not really round.
So is there anyone out there who did better in geometry class than I did who can tell me how close I am?
Thanks
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Warren H.

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Permit me to suggest that's not what you should do. Your raised bed should be of such size and shape that you NEVER have to step on it. In other words, you should be able to reach the center of your raised bed while your feet are on the ground outside the raised bed.
vince norris
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vincent p. norris wrote:

Thanks for the input, but I do plan to have access to the entire bed. I'm planning on two tiers, with access to the inner ring. The surface of the whole outer ring will be accessible from the outside, and the inner ring will be accessible from the access point. The design of the bed isn't the issue. Knowing where that center point should be sited before removing what's planted now on the most critical side is the issue.
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Warren H.

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Warren, I have never been good at math BUT what you might try is putting a stake in the dirt and using a string as a large compass to trace your circle. You can walk the area that way and see exactly where it is going to be.
It would be fast to do it this way and when you get it then just go back and follow the mark with a hose, rope or paint.
Kate
:I have a yard area that I want to put a circular raised be in. The area is : sort of a wedge with the tip chopped-off, except that one of the sides (the : west) is along side a driveway that flares into the area as it approaches : the south side (the street). : : The south side is 10' long. The east side is 31' long. The north side is 30' : long. The west side is two segments, with a very obtuse angle (not quite a : straight line); the north segment on the west side is 14' long, and the : south segment is 10' long. The northwest and southeast corners are : 90-degrees. The northeast corner is a little acute. The southwest corner is : a little obtuse. : : I want to create the biggest circle I can that touches the east, south (the : street), and west (the driveway) sides. (It should not touch the north : side.) : : If things don't add-up right, keep in mind that the measurements were : rounded to the nearest half-foot, and the exact location of the northeast : corner could be off slightly. : : How big should the circle be? : : My guess is that it's going to be around 11' or 12' in diameter. The problem : is that because of what's planted where, and when I can do the work, I'll : need to dig-up, and layout the northeast 2/3 of the circle before I can : clear the southwest 1/3. I can guess at a center point, and sweep a string : along that 2/3 of the northeast side, but I can't sweep it over the : southwest 1/3 to be sure it touches both the south and the west side, but : doesn't go over either. There's only so much I'll be able to fudge the : circle without people noticing it's not really round. : : So is there anyone out there who did better in geometry class than I did who : can tell me how close I am? : : Thanks : : -- : Warren H. : : =========: Disclaimer: My views reflect those of myself, and not my : employer, my friends, nor (as she often tells me) my wife. : Any resemblance to the views of anybody living or dead is : coincidental. No animals were hurt in the writing of this : response -- unless you count my dog who desperately wants : to go outside now. : Have an outdoor project? Get a Black & Decker power tool:: : http://www.holzemville.com/mall/blackanddecker / : : :
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If you run string from diagonal corners, the place where they cross might be a good place for the center. Then measure to the closest side. That would be the radius of the circle.

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Carolyn LeCrone wrote:

That won't work. The circle needs to stay completely within the edges, but needs to touch the south edge (the narrow side of the irregularly shaped area), and won't touch the north edge (the wider side of the area). Your method would only allow me to center a circle in the area.
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SVTKate wrote:

As I said in the message, because of what's planted in the southwest corner, I can't get in to check the arc with a string, and certainly not multiple times as I would only be guessing where the center point should go, and it's unlikely I'd guess right the first time.
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pick a set of reference coordinates. You know the east, south and west are tangent to the circle. equation of line is y-b=m(x-a), circle (x-c)^ 2 + (y-d)^2 = r^2. Solve for c, d, r. Choose the r the fits your criteria.
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Salty Thumb wrote:

Gee. I didn't realize it was that simple.
Nor have I felt as dumb as I do right now looking at that equation. I have no idea where to even start.
One of the things complicating this is that there are two segments on the west side. I'm not sure if the circle will be touching both, or just one.
It's to the point that I'm ready to scrap this project. Solving this problem in a practical way looks like I'm going to have to clear the whole area at once, which isn't something I'll be able to have enough labor to do at the right time of the year in the available window. I could just put in a smaller circle that doesn't touch the edges, and hope it's placement doesn't look too "off", but then I'll have some left-over scrap areas that'll need to be cared for in some way. (It's hard to describe it all, as it is such an irregular space.)
I'm ready to just toss the whole concept of repeating circles, and go with something more free-form. I did think that the repeating circles vs. the odd angles of things I can't move (like the driveway, the house, the street and the lot line) would look good, but I obviously underestimated the difficulty of the geometry when one can't just clear the land, and use more practical methods like stakes, strings, and chalk lines that can be easily erased if drawn wrong.
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Well if you start with y=0 as your south line, it should be easier and you should be able to find m the slope and (a,b) the offset for the other two/three lines by measuring from your reference point. Then plug into your circle equation to solve. You also have your r approximation, you can interatively go through possibilities till you find an acceptable c, d. Should be pretty easy with a spreadsheet. If you have access to Mathmatica or Mat(h)lab or maybe even some Internet applet you could probably just type in the equations to get an answer.. As far as the west side, I just assumed they were close enough to be straight using the southwest line, but if you don't get an acceptable answer, you could always try again with the northwest line. It is also possible that I am off my nut, and you won't be able to get an answer with just those equations.

There might be some easy theorem that would solve you easily, but the hell if I could tell you what it is. As for practical methods, there might be a way, but again, I couldn't tell you how. There must be some Pyramid or Stonehenge builders hanging out in other groups.
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How about this:
Draw it to scale on graph paper, then use a compass!
You can then transfer the layout from paper to planter with very little stress.
Kate
: Salty Thumb wrote: : > : > pick a set of reference coordinates. You know the east, south and west : > are tangent to the circle. equation of line is y-b=m(x-a), circle (x-c)^ : > 2 + (y-d)^2 = r^2. Solve for c, d, r. Choose the r the fits your : > criteria. : > : : Gee. I didn't realize it was that simple. : : Nor have I felt as dumb as I do right now looking at that equation. I have : no idea where to even start. : : One of the things complicating this is that there are two segments on the : west side. I'm not sure if the circle will be touching both, or just one. : : It's to the point that I'm ready to scrap this project. Solving this problem : in a practical way looks like I'm going to have to clear the whole area at : once, which isn't something I'll be able to have enough labor to do at the : right time of the year in the available window. I could just put in a : smaller circle that doesn't touch the edges, and hope it's placement doesn't : look too "off", but then I'll have some left-over scrap areas that'll need : to be cared for in some way. (It's hard to describe it all, as it is such an : irregular space.) : : I'm ready to just toss the whole concept of repeating circles, and go with : something more free-form. I did think that the repeating circles vs. the odd : angles of things I can't move (like the driveway, the house, the street and : the lot line) would look good, but I obviously underestimated the difficulty : of the geometry when one can't just clear the land, and use more practical : methods like stakes, strings, and chalk lines that can be easily erased if : drawn wrong. : : -- : Warren H. : : =========: Disclaimer: My views reflect those of myself, and not my : employer, my friends, nor (as she often tells me) my wife. : Any resemblance to the views of anybody living or dead is : coincidental. No animals were hurt in the writing of this : response -- unless you count my dog who desperately wants : to go outside now. : Have an outdoor project? Get a Black & Decker power tool:: : http://www.holzemville.com/mall/blackanddecker / : : :
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Warren wrote:

I didn't do real good at geometry class, either, but I think I did come up with a reasonable scale drawing of the space you described. If I'm close, then a 12' diameter circle will NOT be big enough to do what you want it to. The center of such a circle would need to be 6' up from the street and 6' in from the driveway if it's going to touch both, and it's not going to touch the west/northwest "almost straight" boundary at all.
Instead, you're going to need something closer to 17-18' in diameter (about 8.5' in radius) centered 8.5-9' north of the street edge and 8.5-9' west of the driveway edge.
I'd suggest you draw out your own scale drawing, though--not too hard to do if you cut out line segments of the appropriate length, lay down the easy parts first (south and east side) and then test position the other three segments until you get a mostly right angle in the northwest.
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DrLith wrote:

I think you're right. I cut-out some paper strips to represent the sides. Took me a good half-hour to get them down right, and that was even after using corners of the paper to make combined south and east, and north and northwest sides. It took awhile, including re-taping a couple of times when corners didn't line-up, but I finally got it down.
I then used variously sized lids from jars and bottles until I found what that fit the way I was hoping, and when I measured it, it was about 17 units in diameter.
Truth is, I never did get it to fit as snugly into the narrow end as I was seeing when I was looking at the actual location. This leaves me with two too big areas in the southwest and southeast corners, and a circular area far bigger than I want. I can't believe my perception from the ground was so far off.
Even if I fudge everything in favor of what I envisioned, I still don't get down to anything close to the 12' diameter I was guessing.
Of course this is exactly why I wanted to know all my measurements *before* I started digging, and placing stones.
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Warren H.

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Making a paper model sounds like a good idea to me, but if you want to work on the ground instead, bisect the southeast angle and bisect the southwest angle. Where the bisectors meet is the center of your circle. Note that because your west side is bent, there may not be a perfect circle that touches all 3 sides, but the bisection method will find it if it exists. .

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I want to add that measuring the sides doesn't give you enough information to make an accurate scale model. You must also measure the angles (very difficult to be accurate) or the diagonals (recommended), and ensure that your model is accurate those ways too.

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I thought this is what Caroline suggested but apparently not. The radius, r, of the circle to the south end should form the same edge for triangles made from half of each bisected angle. By symmetry, the west and east side should have the same r, giving you a correct solution. Good job.
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I would probably draw as accurate a map as possible, decide about where to put the circle, go out and place a stake on each side of where I figured it would fit, measure the distance between the stakes, divide that in half and at the halfway mark put another stake, tie a long string at least enough to reach one of the other stakes, then use that as a protractor and draw the circle. If it is going over a property line on one of the other sides, back all three stakes down the needed distance and redraw. you may have to make the diameter smaller in order to stay inside your perameters. respectfully, lee
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I would just walk around and put about 10 stakes in the ground by eye. Then string a couple strings across to find the center and then mark your circle from there. You might have to do it 2 or 3 times. Good Luck
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