Everything Follows From the Radius
A circle has one parameter. Given any of the four common measurements, the other three follow immediately:
| Property | Formula | From the area |
|---|---|---|
| Radius | r | √(A/π) |
| Diameter | 2r | 2√(A/π) |
| Circumference | 2πr | 2√(πA) |
| Area | πr² | A |
Circle Chart
| Radius | Diameter | Circumference | Area |
|---|---|---|---|
| 1 | 2 | 6.283 | 3.142 |
| 2 | 4 | 12.566 | 12.566 |
| 3 | 6 | 18.850 | 28.274 |
| 5 | 10 | 31.416 | 78.540 |
| 10 | 20 | 62.832 | 314.159 |
| 20 | 40 | 125.664 | 1,256.637 |
| 50 | 100 | 314.159 | 7,853.982 |
Note the coincidence at r = 2, where circumference and area are numerically equal. It is a quirk of the units, not a property of circles.
Pi
π is the ratio of any circle's circumference to its diameter — the same value for every circle, which is what makes it a constant worth naming. It is irrational, so its decimal expansion never terminates or repeats, and transcendental, which is why squaring the circle with compass and straightedge is impossible.
For practical work, 3.14159 is accurate to five decimal places. NASA uses 15 digits for interplanetary navigation; 39 digits would compute the circumference of the observable universe to within the width of a hydrogen atom.
Arcs, Sectors and Segments
- Arc — part of the circumference: 2πr × angle/360.
- Sector — the pie slice between two radii: πr² × angle/360.
- Chord — the straight line between the arc's endpoints: 2r sin(θ/2).
- Segment — the region between a chord and its arc: ½r²(θ − sin θ), with θ in radians.
Radians
One radian is the angle subtending an arc equal to the radius, making a full circle 2π radians. Radians are used throughout calculus and physics because they make the derivative of sine equal cosine without a conversion factor — a simplification that degrees would spoil.
Frequently Asked Questions
Why is the area πr²?
Cut the circle into many thin sectors and rearrange them alternately into a near-rectangle. Its height is r and its width approaches half the circumference, πr — giving πr² in the limit.
How much bigger is a 16-inch pizza than a 12-inch?
Areas of 201 and 113 square inches, so 78% more food for what is usually a much smaller price difference. Diameter comparisons systematically mislead.
What is the difference between a circle and a disc?
Strictly, the circle is the curve itself and the disc is the region enclosed. In ordinary usage "area of a circle" means the area of the disc.