The circle
If you know one measurement of a circle, you know them all. Enter the one you have and get the rest — plus the arc and the area of a sector.
The explanation is shown. Quick: the whole circle and the ring. Advanced: also chord and rise, sectors — and tables for other measurements.
Your numbers
More options: circular sector
Answer
The radius
5cm
Diameter
10cm
Circumference
31.42cm
Area
78.54cm²
Arc length (90°)
7.85cm
Area of sector (90°)
19.63cm²
How to work it out
- 1r = d / 2 = 10 / 2 = 5 cmThe radius is half the diameter — from the centre out to the edge.
- 2O = 2 · π · r = 31.42 cm
- 3A = π · r² = π · 5² = 78.54 cm²Only the radius is squared — not π. A typical slip is (π · r)².
- 4Arc = 90° / 360° · 31.42 = 7.85 cm
- 5Crop = 90° / 360° · 78.54 = 19.63 cm²
The formula
d = 2 · r
The diameter is twice the radius.
O = π · d = 2 · π · r
π (pi) ≈ 3.14159 — the ratio between circumference and diameter in every circle.
A = π · r²
arc = v/360 · C · sector = v/360 · A
A sector with the angle v is v/360 of the whole circle.
In depth: more numbers and “what if”
What if the radius is multiplied by k?
| k | The radius | Circumference | Area |
|---|---|---|---|
| 0.5 | 2.5 cm | 15.71 cm | 19.63 cm² |
| 1 | 5 cm | 31.42 cm | 78.54 cm² |
| 2 | 10 cm | 62.83 cm | 314.2 cm² |
| 3 | 15 cm | 94.25 cm | 706.9 cm² |
The circumference follows the radius, but the area grows with k²: double the radius gives four times the area.
Sectors with other angles
| Angle | Arc length | Area of sector |
|---|---|---|
| 30° | 2.62 cm | 6.54 cm² |
| 45° | 3.93 cm | 9.82 cm² |
| 60° | 5.24 cm | 13.09 cm² |
| 90° | 7.85 cm | 19.63 cm² |
| 120° | 10.47 cm | 26.18 cm² |
| 180° | 15.71 cm | 39.27 cm² |
| 270° | 23.56 cm | 58.9 cm² |
More: Geometry
- PythagorasFind the missing side, set out a right angle, check whether a room is square, and find the space diagonal.
- Area and perimeterRectangle, triangle, trapezium, circle and composite shapes — area and perimeter.
- Volume and surface areaBox, cylinder, cone, sphere, pyramid, prism, pipe and frustum — also in litres.
- Right triangleSine, cosine and tangent: all sides and angles from two known values.
- AnglesThe third angle in a triangle and the angle sum of a polygon.
- Points in the coordinate systemDistance, midpoint and the line between two points — drawn in.
- Similar figuresScale factor and missing sides — area is multiplied by k², volume by k³.
- Any triangleThe sine and cosine rules: find the rest from SSS, SAS, ASA or SSA.
- TransformationsReflect, rotate and translate a figure on a coordinate grid — before and after.