What the Polsby-Popper score actually measures

If you've read three articles about gerrymandering, you've probably read the word "compactness." It gets used like a verdict: compact districts are good, non-compact districts are gerrymandered. The implication is that there's an objective geometric test, and the districts that fail it are the problem.

That's mostly right. It's also incomplete in ways that matter.

The formula, in one line

Polsby-Popper is the most common compactness score in U.S. redistricting law. It comes from a 1991 paper by Daniel Polsby and Robert Popper. The formula is:

PP = (4 × π × Area) / (Perimeter²)

That's it. Take the area of the district. Multiply by 4π. Divide by the perimeter squared.

The result is always between 0 and 1.

• A perfect circle scores exactly 1.0. Same area-to-perimeter ratio you could ever achieve.
• A square scores about 0.785.
• A long, narrow rectangle scores well below 0.5.
• A district that wraps around several counties to pick up specific voters can score below 0.1.

The intuition is simple. A compact shape minimizes its perimeter for a given area. The more squiggly the boundary, the more perimeter, the lower the score. If a district has a long, finger-shaped extension that wraps around a neighborhood, that finger adds a lot of perimeter and almost no area. The score crashes.

Two examples

Look at Illinois's 4th Congressional District, often called the "earmuff district" because of the two lobes of territory connected by a thin strip running along Interstate 294. Using current TIGER/Line boundaries, IL-4 scores a Polsby-Popper of 0.1193. That's very low. The shape has been used as a textbook illustration of "what gerrymandering looks like" for more than thirty years.

Now look at Wyoming's at-large district. It's the entire state. Wyoming is roughly rectangular, with straight survey-line borders. Its Polsby-Popper score is 0.7265, near the upper end of what any real-world district achieves. There's no clever redistricting happening here, because there's nothing to redistrict: Wyoming gets one seat, and the district is the state.

Those two examples give you the intuition. They also set up an important caveat that we'll come back to: IL-4 was drawn that way on purpose, and not for the reason most people assume.

Where compactness gets it right

In states like North Carolina, Pennsylvania, and Maryland, low Polsby-Popper scores have repeatedly tracked with documented partisan intent. The 2018 Pennsylvania Supreme Court decision that threw out the state's congressional map cited compactness, among other factors. The math worked. When a district's shape can only be explained by chasing or avoiding specific groups of voters, the geometric signature is real, and Polsby-Popper catches it.

For most "you'll know it when you see it" gerrymanders, the formula gives you a number that confirms what your eyes told you.

Where compactness gets it wrong

Now the harder part.

Geography compresses scores. Districts in mountainous, coastal, or island-heavy areas pick up perimeter from rivers, ridges, and shorelines they don't control. A district along the Mississippi River will look less compact than a district in flat Iowa, and not because anyone gerrymandered it. Hawaii's geography alone makes it nearly impossible to draw compact districts.

Communities of interest don't have round borders. A district that follows the boundary of a Native American reservation, or that consolidates a metropolitan area that happens to be shaped like a horseshoe, will score poorly on Polsby-Popper even when its shape is doing exactly the right thing for the people inside it. The 1965 Voting Rights Act sometimes requires the drawing of majority-minority districts whose shapes are dictated by where the relevant community actually lives, not by geometry. Illinois CD-4 from our example above is exactly this case: its earmuff shape exists because the court approved a map that connected two separated Hispanic neighborhoods on Chicago's near-northwest and near-southwest sides to give them combined voting power under the VRA. The geometry looks gerrymandered, and by Polsby-Popper it is. The reason it looks that way is civil-rights compliance, not partisan engineering.

A compact district can still be a gerrymander. This is the one that gets overlooked. A racial or partisan gerrymander doesn't have to look weird. You can draw a perfectly square district that splits a city center exactly down a partisan dividing line and produces a guaranteed result, and Polsby-Popper will tell you it scored 0.785, well within normal range. Compactness measures only one signature of gerrymandering. It misses several others entirely.

This is why political scientists use Polsby-Popper alongside other measures: the efficiency gap (Stephanopoulos and McGhee, 2015), partisan symmetry, mean-median difference, and Reock (a different geometric ratio based on the smallest enclosing circle). No single number tells you whether a district is fair. They're a panel, not a verdict.

How VoteValue uses it

In VoteValue, Polsby-Popper feeds into the District Integrity subscore of the VVI, alongside the efficiency gap. That gives you the pure geometric signal.

But because compactness alone is incomplete, we also run a separate, human-driven measurement: the Gerrymander Guesser. Users compare real district pairs and rank them by how gerrymandered they look. Every pairwise outcome updates a per-district Perception ELO rating, using the same algorithm as competitive chess ratings. A district that the public consistently flags as gerrymandered will climb in ELO, even if its Polsby-Popper score is only moderately low, because human judgment captures contextual cues, like an obviously partisan boundary that cuts a town in half, that geometry alone can't see.

Compactness math and crowd judgment, side by side. Neither one tells the whole story, but together they tell more than either does alone.

The bigger lesson

Compactness is a great first question. It's a terrible last question.

If someone tells you a district is gerrymandered "because look at the shape," they might be right. But if they only ever invoke the shape, they're underselling the case. The shape is one of several geometric fingerprints, and not even the most damning one. The damning one is usually what the district does, not what it looks like: how reliably it produces an outcome that doesn't match the voters inside it.

That's the harder thing to see, and the more important one.

If you want to see your own district's Polsby-Popper score, type any U.S. address into VoteValue and expand the District Integrity card. The map page also has a "Compactness" color mode that shades every district in your state by its score.