Math101 · Independent Ontario learning library
Math101learn.math101.caArea 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.
Worked example
Representations and interpretation
Cutting a disk into many narrow sectors and alternating them creates a shape approaching a rectangle with base $\pi r$ and height $r$, explaining $A\approx(\pi r)r=\pi r^2$.
Reasoning about variations
Doubling diameter also doubles radius and quadruples area. For an annulus, subtract inner disk area from outer: $\pi R^2-\pi r^2=\pi(R^2-r^2)$.
Common mistakes
How to check your work
- Compare area with the containing square of side $2r$: it should be below $4r^2$.
- Estimate $\pi$ as a little over $3$.
- Reverse using $r=\sqrt{A/\pi}$ when appropriate.
Practice
- Find the area of a circle with radius $6$ cm.
- Find the area when diameter is $10$ m.
- By what factor does area change if radius triples?
Answers and brief solutions
Show answers
- $36\pi$ cm$^2$ $A=\pi(6)^2$.
- $25\pi$ m$^2$ Radius is $5$ m.
- $9$ Area depends on $r^2$.
Synthesis and transfer
Comparing two circular garden beds through radius ratio predicts their area ratio before either exact area is computed, and units must finish as square units.
If one garden has radius $3$ m and another radius $6$ m, their radius ratio is $2$ but their area ratio is $2^2=4$. The larger area is $36\pi$ square metres, not twice $9\pi$. This quadratic scaling explains why a modest increase in pizza diameter produces a much larger increase in food. Solving backward from area requires the principal square root because radius is nonnegative. An estimate using $\pi\approx3.14$ can assess decimal plausibility, while leaving $\pi$ in the exact result preserves the circular relationship until a measurement precision is specified.
Related topics
Teaching and accessibility note
Explore the idea
Geometry measurement
Change one quantity at a time and connect what moves to Area of a Circle.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the area of a circle with radius $6$ cm.
- $A=\pi(6)^2$.
End of lesson
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