# GeographyNumerically defining parcel irregularity

#### UrbaneSprawler

##### Cyburbian
I suspect most zoning codes recognize compliance with development code requirements is inherently more difficult when a parcel of land is not shaped like a square or a rectangle. In our neck of the woods, it's becoming more and more that we're seeing development proposals on remnant parcels, and just in general irregularly shaped parcels that look less like Wyoming and more like Idaho.

I'm wondering if there's a tool out there being used to measure irregularity, basically how much deviation a parcel is from a square or rectangle that takes into account the number of sides, how far of a deviation of angle from 90 degrees, etc.? I seems like there can be a quantitative way to measure this, a correlation coefficient of sorts?

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#### mendelman

##### Unfrozen Caveman Planner
Staff member
Moderator
I'm wondering if there's a tool out there being used to measure irregularity, basically how much deviation a parcel is from a square or rectangle that takes into account the number of sides, how far of a deviation of angle from 90 degrees, etc.? I seems like there can be a quantitative way to measure this, a correlation coefficient of sorts?
I'd don't know of such a tool, but the more important question is: Why are you asking?

Is there a real world situation you're trying to overcome? What you describe in your first paragraph is a 'condition', but it doesn't really express why you're asking.

Give us a little more background.

#### JNA

##### Cyburbian Plus
Why hung up on shape over instead of

you say "remnant parcels " - any local regs on number of splits allowed ? or remainders ?
require a survey - is there a minimum lot size ?
frontage/ access
is septic field an issue - is there enough sq ft for a second field bed

#### UrbaneSprawler

##### Cyburbian
My question It's specific to a project, just more the condition perhaps of the build-out of a city, where the greenfield parcels have developed, and the remaining parcels of land are shaped in a manner that trying to get the typical requirements of our code (grid pattern of streets, blocks sizes, access spacing of streets) is inherently a little more challenging when a parcel of land is less rectangular.

Certainly there are additional considerations on a particular parcel of land, topography, natural features, utilities, etc. But all things being equal otherwise and there are two parcels of land, one shaped like Wyoming, the other like Maryland, meeting aspects of our development code is more difficult to develop a pattern of streets, blocks, etc.

I'm just wondering if anyone's ever tried to quantitatively define irregularity, rather than the Justice Potter "I know it when I see it." approach, which does work in my view. It would be interesting to give a metric to it to our board though.

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#### estromberg

##### Cyburbian
This sort of analysis can be done with GIS. You'd have to set up some criteria to test and index like number of vertices with deflections of more than say 20 degrees, area to perimeter ratio, etc. You might want to have a look at some of the analysis of political districts for gerrymandering indexing as there are definitely some similarities.

#### mendelman

##### Unfrozen Caveman Planner
Staff member
Moderator
Really it sounds like you're trying to shove a square peg (the irregular parcels) into a round hole (your, seemingly, inflexible code).

I think you're looking for a solution to the wrong problem.

Good luck.

#### Faust_Motel

##### Cyburbian
I'd just go for a shape factor analysis. Lots of places have shape factor requirements for new parcels. If what you're trying to do is figure out how big or how extensive your "existing irregular parcel problem" is, this could be a quick way to do it.

I'm not much of a GIS technician at this point, but as I understand it shape factor might compare the area of the smallest rectangle you can draw around a parcel to the area of the actual parcel. The bigger the gap between those two, the more irregular the shape.

#### estromberg

##### Cyburbian
The term you'll want to look at for GIS is shape index and minimum bounding rectangle.

#### UrbaneSprawler

##### Cyburbian
estromberg, thanks a bunch! The PSI example is exactly what I was envisioning, the example on page 7 in particular hits this perfectly. Super appreciate this information!

The term you'll want to look at for GIS is shape index and minimum bounding rectangle.

#### UrbaneSprawler

##### Cyburbian
This sort of analysis can be done with GIS. You'd have to set up some criteria to test and index like number of vertices with deflections of more than say 20 degrees, area to perimeter ratio, etc. You might want to have a look at some of the analysis of political districts for gerrymandering indexing as there are definitely some similarities.
estromberg, thank you again for this as well. The Parcel Shape Index article was fascinating but the maths I suspect for folks to quickly derive at a calculation doesn't seem to be possible. It seems like some sort of GIS plug-in would be needed. But, when you mentioned gerrymandering indexing as a suggestion, I started looking into that and came across the Polby-Popper ratio. It seems to potentially work for our discussion purposes. A circle has a ratio of 1, a rectangle looks to be about .7, and an irregular parcel I did a calc on measured .29. It might be a cool tool to try. Thanks so much!

#### Dan

Staff member
Moderator
Shape factor! I had to look for it in an older draft of the FBC I've been working on. The powers that be rejected the idea, because just the idea of adopting an FBC was challenging enough, but anyhow ....

P² / A = SF

P = lot perimeter (any unit of length - meters, feet, northern New Mexico vara, etc.)
A = lot gross area (same unit of length as P)
SF = shape factor

Example: a certain rectangular lot is 100’ wide and 200’ deep. Lot perimeter is 600’ (100+100+200+200=600), and lot area is 20,000 ft² (100 × 200 = 20,000). Lot perimeter squared is 360,000 (600 × 600 = 360,000). 360,000 ÷ 20,000 = shape factor of 18.

The code called for a shape factor of ≤ 21.3 for a T-3 lot, and ≤25.0 for T-4 and T-5.

#### UrbaneSprawler

##### Cyburbian
Dan, thanks for adding Shape Factor to the mix! I added that to a spreadsheet with Polsby-Popper and it's been interesting comparing the data with parcels in our area. I think I would tend to prefer Polsby-Popper as it always has a value less than 1 (unless the parcel is a perfect circle). Rectangular parcels are pretty much .7 (a square is .8). The odd parcel I'm using as a test case has a Polsby-Popper of .29, and a Shape Factor of 42.80.

I'd be curious if you wanted to try comparing Polsby-Popper vs. Shape Factor? The formula for it is below:

Maybe it would gain more traction with your FBC challenge?

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Dan

#### Dan

Staff member
Moderator
I like the idea of Polsby-Popper (first time I ever heard of it), with "1" being a perfectly compact shape. I ran into problems for two reasons, really. First, the idea of regulating lot shape beyond simple minimum/maximum area and minimum/maximum width at the building setback line seemed "excessive" (even though we have a lot of flag, fan, and other oddball lots in the community). Second, "math hard." Throwing π in there might not help, but at least it's not e or φ. (I think φ is the perfect lot shape, for what it's worth.)

I was searching for a formula for parcel shape compactness a while back, and found a bunch of subdivision codes in Massachusetts that use shape factor. Googling around, it seems like many set a maximum shape factor of 22, 30, or 22 up to [x] sf, 30 above.

Without shape factor, I had to use qualitative guidelines, not a quantitative regulation, to promote lot regularity.
• A lot should have a compact shape, as close to a square, rectangle, or isosceles trapezoid as possible.
• Lot lines at the side (corner side, interior side, side alley) should be as perpendicular or radial to the fronting thoroughfare as possible.
• Lot lines at the rear (rear, rear alley) should be as parallel to the fronting thoroughfare as possible.

#### UrbaneSprawler

##### Cyburbian
Dan, I'm curious to understand more the intention of using shape factor values as a requirement for the different transects. I was originally thinking this meant that the overall parcel (comprised of several lots) say in T-3 had to have as parcel shape ≤ 21.3. But I'm reading this now that the individual lots inside the overall parcel had to meet this metric. I tend to always see lots being rectangular and at right angles to the street. Are developers looking to create non-rectangular lots on purpose?

#### Dan

Staff member
Moderator
I found an interesting formula that Shrewsbury, Massachusetts uses for shape factor: The linked file also explains why regulating shape can be justified, beyond simple buildability.

(16 x A) / P² = SF

P = lot perimeter (any unit of length)
A = lot gross area (same unit of length as P)
SF = shape factor

With this formula, a perfectly square lot has a shape factor of 1. SF for a 2:1 rectangular lot is 0.88, for 3:1 it's 0.75, 4:1 it's 0.64, and 5:1 it's 0.55. Lot shapes start to get weird or unusually narrow around SF 0.6 or below.

#### UrbaneSprawler

##### Cyburbian
Dan, I spent some time a couple weeks ago making a table to compare Polsby-Popper to Shape Factor and used the data of actual parcels in town. Since you posted the formula from Shrewsbury, it was easy to add another column as well. The nice part in any of these is that the numbers keep their pattern throughout. Shrewsbury and Polsby-Popper are very similar. In looking at the actual parcels, I like how with Polsby-Popper, anything below .5 does seem to be irregular (.8 is a square, .7 is a rectangle).

 Perimeter (feet) Area (acres) Polsby-Popper Shape Factor Shrewsbury, MA SF 9,922 52.7 0.29 42.80 0.37 15,994 146.08 0.31 40.12 0.40 7,555 34.85 0.33 37.52 0.43 3,154 6.27 0.35 36.35 0.44 1,295 1.17 0.38 32.84 0.49 2,025 3.14 0.42 29.92 0.53 14,894 180.69 0.45 28.13 0.57 8,053 54.05 0.46 27.49 0.58 6,045 32.3 0.48 25.92 0.62 8,623 65.93 0.49 25.84 0.62 8,722 67.67 0.49 25.75 0.62 30,810 901.73 0.52 24.12 0.66 5,477 31.18 0.57 22.04 0.73 4,242 19.81 0.60 20.81 0.77 8,155 75.78 0.62 20.11 0.80 16,709 358.85 0.70 17.82 0.90 1,909 4.81 0.72 17.36 0.92 322 0.14 0.74 16.97 0.94 10,703 162.29 0.78 16.17 0.99 5,280 40 0.79 15.97 1.00

Dan