SME
Cartography · Problem type
Proportional-symbol radius
Use this for “The largest city, 1,000,000 people, gets a 20 mm circle. How big a circle does a city of 250,000 get?”
Diagram
r
=
rₘₐₓ
√(
v
vₘₐₓ
)
r
=
20
mm
×
√(
250,000
1,000,000
)
=
10
mm
Each circle’s area, not its radius, is in proportion to its value: a place a quarter the size gets a circle half as wide.
This is an example. Type your own values to start a new problem.
r (mm)
=
rₘₐₓ (mm)
v
vₘₐₓ
q
Share of the largest value
worked out (not typed)
Formulas
q
=
v
vₘₐₓ
0.25
=
250,000
1,000,000
r
=
rₘₐₓ
×
√
q
10
=
20
×
√
0.25
Assumptions
•
The circle’s area, not its radius, stands for the value, so the radius grows as the square root of the value.
•
A quarter of the value gets a quarter of the area: half the radius.
•
Readers tend to underjudge big circles; some maps enlarge them a little (Flannery’s scaling), which this page leaves out.
Step-by-step
WE KNOW
vₘₐₓ = 1,000,000, rₘₐₓ = 20 mm, v = 250,000
FIND
share of the largest value (q), symbol radius (r)
Step 1 · Find share of the largest value (q)
Use q = v ÷ vₘₐₓ
The value as a share of the largest.
q
=
250,000
1,000,000
q
=
0.25
Step 2 · Find symbol radius (r)
Use r = rₘₐₓ × √(q)
Areas
keep
the
share
q
,
so
radii
keep
its
square
root.
r
=
20
×
√
0.25
√
0.25
=
0.5
r
=
10
mm
Check
Put the numbers back in. Both sides should match.
0.25
=
250,000
1,000,000
✓
10
=
20
×
√
0.25
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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