A circle has four quantities people almost always ask about: radius, diameter, circumference and area. The useful thing is that as soon as you know one of them, you can work out all the others. This guide shows the formulas, the working for every case and typical examples from the kitchen, garden and workshop. After that it covers the composite shapes – annulus, sector and segment – and the mistakes that happen most often when calculating with circles.
The parts of a circle
A circle is the set of all points at the same distance from a centre point M. That distance is the radius. Every other term builds on it:
- Radius r: distance from the centre to the edge.
- Diameter d: distance from edge to edge through the centre, twice the radius.
- Circumference C: the length of the circle's boundary, once all the way round.
- Area A: the region enclosed by the circle.
- Chord s: a straight line joining two points on the circle. The diameter is the longest chord.
- Arc b: the part of the circle between two points.
- Central angle α: the angle between two radii at the centre.
The basic formulas
Every circle formula is based on the number π (pi), roughly 3.14159. It tells you how many times the diameter fits into the circumference – the same for every circle. Two formulas follow from that, and they are worth memorising:
The circumference grows in proportion to the radius, the area with its square. Double the radius and the circumference doubles, but the area quadruples. More on that in the article Area of a circle.
Working out everything from one value
The safest route always goes through the radius: work out r first, then put it into the basic formulas. This table shows how to get the radius from each known quantity:
| Known | Radius | Example |
|---|---|---|
| Diameter d | r = d ÷ 2 | d = 30 cm → r = 15 cm |
| Circumference C | r = C ÷ (2 × π) | C = 157 cm → r = 24.99 cm |
| Area A | r = √(A ÷ π) | A = 4 m² → r = 1.128 m |
Example 1: radius given
A circle has a radius of r = 5 cm. The diameter is d = 2 × 5 = 10 cm. The circumference is C = 2 × π × 5 = 31.42 cm and the area A = π × 5² = π × 25 = 78.54 cm².
Example 2: diameter given
A round dining table is 1.20 m across, so the radius is 0.60 m. For a tablecloth you need the circumference: C = π × 1.20 = 3.77 m. The table top has A = π × 0.60² = 1.131 m². As a rule of thumb you can also use d² × 0.785 for the area, because π ÷ 4 ≈ 0.785.
Example 3: circumference given
The diameter of a tree trunk is hard to measure directly, but its girth is easy with a tape measure. With a circumference of 157 cm:
The diameter is therefore just under 50 cm and the cross-section π × 24.99² = 1,961.5 cm². If you need the area straight from the circumference, use:
More on this direction in the article Circumference of a circle.
Example 4: area given
A round flower bed should be exactly 4 m². What radius does it need? Divide the area by π and take the square root:
The diameter is 2.257 m, and the edging needs C = 2 × π × 1.128 = 7.09 m.
Common diameters at a glance
| Diameter | Circumference | Area | Typical for |
|---|---|---|---|
| 10 cm | 31.42 cm | 78.54 cm² | coaster |
| 20 cm | 62.83 cm | 314.16 cm² | small pizza, plate |
| 26 cm | 81.68 cm | 530.93 cm² | springform cake tin |
| 30 cm | 94.25 cm | 706.86 cm² | large pizza |
| 50 cm | 157.08 cm | 1,963.50 cm² | tree trunk, planter |
| 100 cm | 314.16 cm | 7,853.98 cm² | table, paddling pool |
The annulus (ring)
An annulus is the area between two circles with the same centre – the cross-section of a pipe, a washer or a path around a flower bed. Its area is the large circle minus the hole:
R is the outer radius and r the inner radius; the ring width is R − r. Example: a pipe has an outer diameter of 110 mm and a wall thickness of 3.2 mm, so R = 55 mm and r = 51.8 mm:
That is how much material goes into every millimetre of pipe. A common mistake is to multiply the ring width by the outer circumference, which gives too much. The correct length is the mean circumference: π × (R + r) × (R − r) gives exactly the same result as the formula above.
The sector
A sector is bounded by two radii and the arc between them – like a slice of cake. It is simply a fraction of the full circle, namely α ÷ 360°:
Example: a cake from a 26 cm tin is cut into 12 equal slices. Each slice has an angle of 360° ÷ 12 = 30° and a radius of 13 cm. The arc at the edge is 2 × π × 13 ÷ 12 = 6.81 cm, the area π × 13² ÷ 12 = 44.24 cm². How to convert angles to radians is explained in Sector, arc length and radians.
The segment
A segment is bounded by a chord and the arc. It is the shape of an arch above a door or a shallow vault. You get its area by subtracting the triangle formed by the two radii and the chord from the sector:
Here is the angle in radians. Chord s and arc height h (the rise) relate to radius and angle like this:
In practice you often only know the span and the rise, i.e. chord and height. The radius is then:
Example: an arch has a span of 1.20 m and a rise of 0.30 m. The radius is 1.44 ÷ 2.4 + 0.15 = 0.75 m. The central angle comes out at 106.26°, or 1.8546 in radians. The arch area is 0.75² ÷ 2 × (1.8546 − 0.96) = 0.2516 m² and the arc length 0.75 × 1.8546 = 1.39 m. In the circle calculator choose “Segment” and “Chord + height”.
Converting units correctly
Lengths and areas convert differently. For lengths the plain factor applies: 1 m = 100 cm. For areas the factor is squared: 1 m² = 100 × 100 cm² = 10,000 cm². To convert a circle area of 7,853.98 cm² into square metres, divide by 10,000 to get 0.785 m², not 78.5 m².
| Area unit | in m² |
|---|---|
| 1 mm² | 0.000001 m² |
| 1 cm² | 0.0001 m² |
| 1 dm² | 0.01 m² |
| 1 a (are) | 100 m² |
| 1 ha (hectare) | 10,000 m² |
| 1 km² | 1,000,000 m² |
The most common mistakes
- Mixing up radius and diameter: putting the diameter into A = π × r² gives four times the area.
- Forgetting the square: π × r is not an area. Only π × r × r gives the area.
- Converting areas linearly: cm² to m² uses the factor 10,000, not 100.
- Rounding π too early: if you use 3.14 in the first step and keep calculating, you carry the error along. Round only the final result.
- Mixing degrees and radians: your calculator must be set to degrees (DEG) for sin α when α is in degrees. In the segment formula, however, the term before the sine is the angle in radians.
Conclusion
One known quantity is enough for any circle: it gives you the radius, and the radius gives you everything else. Annulus, sector and segment are just combinations of the same formulas. If you would rather not do the working yourself, the circle calculator does it for you – with a diagram, step-by-step solution and any number of decimal places.