What a Circle Calculator Computes and How Each Formula Follows
A circle is the set of points at a fixed distance from a centre, and that single distance — the radius — determines everything else about it. Give this calculator any one of the radius, diameter, circumference or area and it returns the other three, then goes further: sector area, arc length, chord length, segment area, sector perimeter and the equation of the circle from a centre you choose. Machinists working out material for a round flange, gardeners edging a circular bed, students checking geometry homework and designers spacing items around a dial all start from whichever measurement they can actually take. Sometimes that is a tape measure around the outside, which gives circumference; sometimes it is a drawing that only lists the area.
Three relations connect the four core values, exactly as calculator.net states them: d = 2r, C = 2πr and A = πr². Here r is the radius, d the diameter, C the circumference and A the area, all in whatever length unit you use — areas come out in that unit squared. Working backwards is algebra: r = C / (2π), or r = √(A / π) from an area. For a slice, with θ the central angle, the sector area is ½r²θ and the arc length is rθ when θ is in radians, which is why the conversion θ_rad = θ_deg × π / 180 happens first. The chord is 2r·sin(θ/2), and the segment — the region between chord and arc — is ½r²(θ − sin θ), the sector minus the triangle inside it. The circle's equation is (x − h)² + (y − k)² = r² for a centre (h, k).
Take a radius of 5. The diameter is 10, the circumference is 31.415927 and the area is 78.539816. Add a central angle of 60° and the slice values follow: θ becomes 1.047198 radians, the arc is 5 × 1.047198 = 5.235988, the sector area is 0.5 × 25 × 1.047198 = 13.089969, and the chord is 2 × 5 × sin(30°) = 5 exactly — the chord equals the radius at 60°, because the two radii and the chord form an equilateral triangle. The segment area is 0.5 × 25 × (1.047198 − 0.866025) = 2.264652. Reverse the problem and it still holds: enter an area of 50 and the calculator returns r = 3.989423, d = 7.978846 and C = 25.066283. Place that radius-5 circle at centre (2, −3) and the equation reads (x − 2)² + (y + 3)² = 25, or x² + y² − 4x + 6y − 12 = 0.
The sector and segment figures are where this tool earns its place. Sheet-metal and upholstery work needs the segment area to estimate offcut waste when a curved piece is trimmed by a straight edge. A landscaper pricing a curved patio needs sector area for the paving and sector perimeter — arc plus two radii — for the edging, and those are different numbers. Anyone laying out bolt holes or clock markings needs the chord to set a divider between adjacent positions. And a student solving a mensuration question usually needs the minor segment specifically, which is why the tool labels whether the segment shown is the minor one below 180° or the major one above it.
One pitfall causes most disagreements between calculators: the value of π. This tool uses JavaScript's full-precision Math.PI, 3.141592653589793. Hand-checking with 3.14 or 22/7 introduces an error of roughly 0.05%, which on a 10-metre-diameter circle is about 15 mm of circumference — enough to matter when cutting material. A second trap is mixing units: enter the radius in centimetres and the area is in square centimetres, so convert before you enter. Watch the angle unit too, since 60 radians is not 60 degrees. A zero or negative radius is rejected with a plain explanation instead of a NaN, and an angle above 360° is refused because it would wrap past the whole circle. All of it is computed in your browser, so nothing you enter leaves your device.
