Math101 · Independent Ontario learning library
Math101Area of a Circle
The area enclosed by a circle of radius $r$ is $A=\pi r^2$. The square on $r$ reflects two-dimensional scaling: multiplying radius by $k$ multiplies area by $k^2$.
Circle area supports design, coverage, probability, cylinders, and optimization. Its quadratic scaling is a key geometric modelling principle.
Intuition and core definition
The area enclosed by a circle of radius $r$ is $A=\pi r^2$. The square on $r$ reflects two-dimensional scaling: multiplying radius by $k$ multiplies area by $k^2$. Radius is half the diameter and must be identified correctly before substitution.
Notation, language, and conditions
$r>0$ is centre-to-circle distance, $d=2r$ is diameter, and $A$ uses square units. Exact answers retain $\pi$; decimal answers state a chosen precision. The formula describes a full disk, not its circumference or a sector.
Why this idea matters
Circle area grows with the square of radius, so doubling a radius multiplies covered surface by four rather than two.
A dependable method
- Determine whether the given length is radius or diameter.
- Convert to radius in consistent units.
- Square the radius before multiplying by $\pi$.
- Report exact and, if requested, approximate area in square units.
- Compare with a circumscribed square or scaling expectation.
