Proportional symbols show a value by the size of a symbol. When circles are scaled by area, the value depends on the square of the radius. This appears in map and data questions, and it belongs to graphs, photographs and spatial data.
How do you read a proportional circle?
- Read the key. Find the reference circle: its radius and its value.
- Check the method. Is the area proportional to the value? Many keys say so, and some give a radius-based rule instead.
- Measure the radius of the circle you need, in the same units as the key.
- Find the radius ratio and square it if area is proportional.
- Multiply the key value by that area ratio.
Worked example
All data is invented. The key says a circle of radius 5 mm represents 100 000 people, with area proportional to population.
City B has a circle of radius 10 mm.
- Radius ratio: 10 ÷ 5 = 2
- Area ratio: 2² = 4
- Population: 4 × 100 000 = 400 000
City C has a circle of radius 7.5 mm.
- Radius ratio: 7.5 ÷ 5 = 1.5
- Area ratio: 1.5² = 2.25
- Population: 2.25 × 100 000 = 225 000
Check the direction: C is bigger than the key circle but smaller than B, and 100 000 < 225 000 < 400 000. The order fits.
What mistake do students make?
Mistaken answer: “City B’s circle is twice as wide, so it has twice the population: 200 000.”
That treats the radius as proportional to the value. The key says area is proportional, and doubling the radius makes the area four times larger. The correct value is 400 000.
The correction is to ask what the key scales, then use the matching ratio. If a key were radius-based, 200 000 would be right, which is why the key must be read first.
Check yourself
Same invented key: 5 mm radius is 100 000 people, area proportional.
1. A circle has radius 15 mm. What population does it show?
Show answer
15 ÷ 5 = 3, and 3² = 9. Population = 9 × 100 000 = 900 000.
2. Two circles have radii 6 mm and 12 mm. How many times larger is the larger value?
Show answer
Radius ratio 12 ÷ 6 = 2. Area ratio 2² = 4 times larger.
3. What radius should a circle have to show 50 000 people?
Show answer
Value ratio = 50 000 ÷ 100 000 = 0.5. Radius = 5 × √0.5 = 5 × 0.707 ≈ 3.5 mm.
Where this leads next
Symbols and graphs both invite you to link patterns, so continue with identifying an unsupported spatial correlation. The statistics and distribution explorer shows how values spread, which helps when judging symbol sizes. A teacher can also check your reading of unfamiliar keys in our online one-to-one Geography tuition.