SME
Human Geography · Problem type
How compact a district’s shape is
Use this for “A voting district has area 1,000 km² and perimeter 200 km. How compact is it?”
Chart
200 km
This is an example. Type your own values to start a new problem.
PP
=
4π
×
A (km²)
÷
(
P (km)
)
2
C
Circle’s perimeter
worked out (not typed) · km
Formulas
PP
=
4π
×
A
P
2
0.3142
=
4π
×
1,000
200
2
C
=
2
×
√
π ×
A
112.1
=
2
×
√
π × 1,000
Assumptions
•
The Polsby–Popper score compares the district’s area with a circle of the same perimeter: a circle scores 1, a square π ÷ 4 = 0.785.
•
A low score flags a stretched or ragged district, one sign of gerrymandering but not proof: coasts and rivers make ragged borders too.
•
The score depends on how finely the border is traced, so compare districts mapped at the same scale.
Step-by-step
WE KNOW
A = 1,000 km², P = 200 km
FIND
Polsby–Popper score (PP), circle’s perimeter (C)
Step 1 · Find Polsby–Popper score (PP)
Use PP = 4π × A ÷ P²
A
circle
with
perimeter
P
holds
P
2
4π
,
so
the
score
is
the
district’s
area
over
that
circle’s.
PP
=
4π
×
1,000
200
2
4π × 1,000 = 12,566
200
2
=
40,000
12,566
40,000
=
0.3142
PP
=
0.3142
Step 2 · Find circle’s perimeter (C)
Use C = 2 × √(π × A)
A
circle
holding
the
same
area
has
radius
√(
A
π
),
so
its
perimeter
is
2π
times
that,
or
2
√
πA
.
C
=
2
×
√
π × 1,000
π × 1,000 = 3,142
√
3
,142
=
56.05
2 × 56.05 = 112.1
C
=
112.1
km
Check
Put the numbers back in. Both sides should match.
0.3142
=
4π
×
1,000
200
2
✓
112.1
=
2
×
√
π × 1,000
✓
Topic check
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