Every measure of a circle from just one
A circle is completely described by a single number: its radius. Once you know it, the diameter, the circumference and the area all follow, and so do the arc and the area of any slice. This calculator starts from whichever measure you have, the radius, diameter, circumference or area, and computes all the others.
The formulas
- Diameter d = 2r.
- Circumference C = 2πr = πd.
- Area A = πr².
- Radius from circumference r = C ÷ 2π, and from area r = √(A ÷ π).
- For a sector of angle θ in degrees: arc length = C × θ ÷ 360, sector area = A × θ ÷ 360 and chord = 2r × sin(θ ÷ 2).
π is about 3.14159. The calculator uses its full precision.
Worked example: a 12-inch pizza
- Diameter 12 in, so radius 6 in.
- Area: π × 6² = 113.10 sq in.
- Circumference: π × 12 = 37.70 in of crust.
- One slice out of six is a 60° sector: arc of 6.28 in, area of 18.85 sq in, and the straight cut across the crust, the chord, is 6 in.
Area by diameter
| Diameter | Area | Circumference |
|---|---|---|
| 8 in | 50.27 sq in | 25.13 in |
| 10 in | 78.54 sq in | 31.42 in |
| 12 in | 113.10 sq in | 37.70 in |
| 14 in | 153.94 sq in | 43.98 in |
| 16 in | 201.06 sq in | 50.27 in |
| 18 in | 254.47 sq in | 56.55 in |
Area grows with the square of the diameter: a 16-inch pizza is only a third wider than a 12-inch one, but it has 78 percent more pizza. Comparing prices per square inch often shows the large size is the better buy.
Everyday uses
- Garden beds and patios: a round bed 10 ft across covers 78.5 sq ft, which tells you how much mulch or how many pavers to buy.
- Fencing or edging around a circular area follows the circumference.
- Pipes and ducts: the cross-section area sets the flow capacity. A 6-inch duct has 2.25 times the area of a 4-inch one.
- Wheels: one turn of a 26-inch bike wheel moves you 81.7 inches forward.
- Baking: comparing round pan sizes is a ratio of areas.
Finding the radius from what you can measure
For a tree trunk or a column, wrap a tape around it to get the circumference, then divide by π to get the diameter: a trunk 94 inches around is about 30 inches across. For a circular area marked on a plan, the area gives the radius via the square root.
Degrees and radians
The sector formulas here use degrees, the unit most people use. In radians, a full turn is 2π instead of 360, and the arc length becomes simply r × θ. One radian is about 57.3°.