How big is a pizza with a 30 cm diameter? How much lawn does a sprinkler with a 6 m reach cover? How much varnish does a round table top need? All these questions come down to one formula: the area of a circle.
The formula: A = πr²
Square the radius and multiply the result by π ≈ 3.14159. A circle with a 6 m radius – roughly the area a sprinkler with a 6 m reach wets – therefore has A = π × 36 = 113.1 m².
Area from the diameter
Usually you know the diameter because it is the easiest to measure. Halve it first, or put it straight into this formula:
A pizza with a 30 cm diameter has π × 900 ÷ 4 = 706.86 cm².
Area from the circumference
If you only know the circumference, for example of a tree trunk or a column:
For a circumference of 157 cm that gives 157² ÷ (4 × π) = 1,961.5 cm².
Why the formula works
Imagine cutting a circle like a cake into a great many thin slices and laying them side by side, alternately tip up and tip down. The result is almost a rectangle. Its height is the radius r and its width half the circumference, π × r. The thinner the slices, the more exact the rectangle – and its area is height times width: r × π × r = π × r².
Double the diameter, four times the area
Because the radius is squared, area grows much faster than diameter. A pizza with a 40 cm diameter is not twice but four times as big as one with 20 cm: 1,256.6 cm² against 314.2 cm².
So when ordering pizza, compare areas rather than diameters. Two 20 cm pizzas have 628.3 cm² between them. A single 30 cm pizza has 706.9 cm² – a good 12% more, even though 30 cm sounds like less than twice 20 cm.
| Diameter | Area | compared with 26 cm |
|---|---|---|
| 26 cm | 530.9 cm² | 100% |
| 28 cm | 615.8 cm² | 116% |
| 30 cm | 706.9 cm² | 133% |
| 32 cm | 804.2 cm² | 151% |
| 36 cm | 1,017.9 cm² | 192% |
Divide the price by the area and you get the price per square centimetre, which lets you compare sizes fairly.
Converting area units
Kitchen circles are measured in cm², garden circles in m². When converting, the length factor is squared: 1 m² is 10,000 cm², 1 cm² is 100 mm². A table top with a 1.20 m diameter has 1.131 m², which is 11,310 cm².
Practical examples
- Sprinkler: 6 m reach, full circle → 113.1 m². Set to a quarter circle in a corner it is 28.3 m².
- Round flower bed: 2 m diameter → 3.14 m². A 5 cm layer of mulch needs 3.14 × 0.05 = 0.157 m³, roughly 160 litres.
- Varnishing a table top: 1.20 m diameter → 1.131 m² per side. At 100 ml per m² and two coats, about 230 ml is enough.
- Scaling a cake recipe: a recipe for a 26 cm tin (530.9 cm²) needs only about 59% of the ingredients in a 20 cm tin (314.2 cm²).
Typical mistakes
The most common mistake is putting the diameter instead of the radius into A = π × r², which makes the result four times too big. Another frequent one is π × r instead of π × r² – that is half the circumference, not an area.
In the circle calculator just enter the radius, diameter or circumference and you get the area instantly in mm², cm², m², ares and hectares. All formulas in context are in the guide Circle formulas.