Area of a Circle Formula
Multiply π by the radius squared.
A = π × r²
How it works
The radius is squared, not the diameter, and that single detail is where most wrong answers come from. You can see why the formula holds by slicing the circle into a great many thin concentric rings and unrolling them: each ring of radius x has length 2πx, and adding them all from 0 to r gives πr². The same result falls out of integrating the circumference with respect to the radius, which is the calculus version of the same picture. Because the radius is squared, area grows far faster than width. Double the radius and the area does not double, it quadruples. A 16-inch pizza is not a third bigger than a 12-inch one: it is (16/12)² = 1.78 times the area, which is why the larger size is almost always better value. π is irrational, so any decimal answer is an approximation; 3.14159 is enough for six significant figures, and engineering work rarely needs more than that because the measured radius will be less precise than π anyway. Watch the units too. Squaring a radius squares its unit, so a radius in centimetres gives an area in square centimetres, and converting afterwards means dividing by 10,000 rather than 100 to reach square metres.
Variables
- AArea
- πPi ≈ 3.14159
- rRadius
Worked examples
| Input | Result |
|---|---|
| r = 1 | π ≈ 3.14159 |
| r = 5 | 25π ≈ 78.54 |
| r = 10 m | 100π ≈ 314.16 m² |
Practical applications
- Sizing pipe, duct and cable cross-sections, where flow capacity depends on area rather than diameter
- Calculating the area of a circular plot, tank base, or centre-pivot irrigation field
- Comparing pizza, cake or tabletop sizes, where the quoted number is a diameter
- Working out how much paint, sealant or turf a circular surface needs
Common mistakes
- Squaring the diameter instead of the radius. That gives four times the correct answer, because d = 2r and (2r)² = 4r².
- Halving the area when you halve the radius. Area scales with the square, so half the radius is a quarter of the area.
- Converting units after squaring rather than before. A radius in centimetres yields square centimetres, and there are 10,000 of those in a square metre, not 100.
- Using 3.14 on work that needs precision. It is low by about 0.05%, which is fine for a patio and not fine for a machined part.
Used in
Use it in a tool
FAQs
What if I only know the diameter?+
Use A = π × (d/2)² = (π × d²) / 4. For d = 10 m, A = 25π ≈ 78.54 m².
How precise should π be?+
Four decimals (3.1416) is enough for engineering. Use more for orbital mechanics.