Circle Calculator
Solve any circle four ways — from a single value (radius, diameter, area or circumference), from three points, from a center and a point on the circle, or from the general equation x² + y² + Dx + Ey + F = 0. Then unlock arc length, chord, sector, circular segment, annulus, tangent and equivalent-square tools.
What is a circle?
A circle is the set of all points in a plane that lie the same distance — the radius — from a fixed point called the center. Because every point on the boundary is exactly one step away from the middle, a circle is the most symmetric shape possible: rotate it by any angle and it looks unchanged.
Circles show up everywhere in the physical world — wheels, coins, plates, pipe cross-sections, orbits, sound waves. Almost anything that spins, rolls or radiates outward is modelled with a circle first.
Circle, part by part
Every circle is fully described by its radius. From that one length you can find the diameter, the distance around it (circumference), the space it covers (area), and every slice or arc you can cut out of it. The cards below show each idea with a small diagram, the formula, and a worked example.
Radius and diameter
The radius is the straight distance from the center of the circle to any point on the edge. The diameter is the full width — one edge, through the center, to the other edge — so it is always twice the radius.
Circumference — distance around
The circumference is the length of the boundary — how far it is to walk once around the circle. It equals 2π times the radius, which is the same as π times the diameter.
Area — space inside
The area is the flat space the circle covers. It grows with the square of the radius, so doubling the radius makes the area four times as big.
Arc and sector — slicing the circle
A sector is a pie-slice bounded by two radii and the arc between them. The arc is the curved edge of that slice. Both depend on the angle θ measured in radians — convert degrees with θrad = θdeg × π/180.
Parts of a circle
- Center: the fixed middle point every boundary point is equidistant from.
- Radius (r): the distance from the center to the edge.
- Diameter (d): a straight line through the center, edge to edge — always d = 2r.
- Circumference (C): the distance once around the edge; C = 2πr = πd.
- Arc: a piece of the circumference between two points on the circle.
- Chord: a straight line joining two points on the circle. A chord through the center is a diameter.
- Secant: a line that crosses the circle at two points (an extended chord).
- Tangent: a line that touches the circle at exactly one point and is perpendicular to the radius at that point.
- Sector: the pie-slice region bounded by two radii and the arc between them.
- Segment: the region between a chord and the arc it cuts off.
Circle formulas
Here r is the radius, d the diameter, C the circumference and A the area, with π ≈ 3.14159…. Every value derives from any other — if you know one of r, d, C or A, the other three are fixed.
How to find each value
About π (pi)
π is the ratio of a circle's circumference to its diameter — the same value for every circle, big or small. Numerically π ≈ 3.14159265…, and its decimal expansion never ends and never repeats: π is irrational, and even stronger, transcendental, meaning it is not the root of any polynomial with rational coefficients. That is why the classical problem of “squaring the circle” with only a compass and straightedge is impossible.
Where circles show up in real life
- Engineering: gears, bearings, pipes, pistons, flywheels — anything that rotates or seals against a round surface.
- Architecture & construction: arches, domes, columns, round windows, silo cross-sections.
- Design & art: logos, badges, radial layouts, dials and typography counters.
- Science: planetary orbits (approximated as circles), lens optics, sound and water waves radiating from a source.
- Everyday life: wheels and tires, pizzas and pies, clocks, coins, plates.
Common conversions
| Quantity | Convert | Value |
|---|---|---|
| Full turn | in radians | 2π ≈ 6.283185 |
| Half turn | in radians | π ≈ 3.141593 |
| Right angle | in radians | π/2 ≈ 1.570796 |
| 1 radian | in degrees | 57.29578 |
| 1 degree | in radians | 0.017453 |
How to use this calculator
- Pick a mode at the top — start with Solve from one value if you already know r, d, C or A.
- Type the value into the matching field. Leave the other three blank; the calculator fills them in.
- Choose a length unit (mm, cm, m, km, in, ft, yd, mi). Area is shown in the matching squared unit plus common alternates.
- For coordinate modes, enter the points exactly as (x, y) pairs — the diagram updates live.
- Click Show calculation steps on any result to see the formula, the substitution, and the final answer.
- Use the arc, chord, sector, segment, annulus and tangent tools below to explore parts of the circle you just solved.
Common mistakes to avoid
- Mixing radius and diameter. Half of the diameter is the radius. Using d in a formula that expects r doubles or halves the answer.
- Forgetting the square in area. A = πr² — square the radius first, then multiply by π. πr on its own is not an area.
- Using degrees where radians are needed. Sector area A = ½ r²θ and arc length s = rθ both need θ in radians. Convert with θrad = θdeg × π/180.
- Chord longer than diameter. A chord can never be longer than 2r. If the input is bigger, the geometry has no real solution.
- Three collinear points. Three points on the same straight line do not define a circle — the circumcircle has infinite radius.
- Tangent from inside. A tangent segment only exists when the external point is outside the circle (d > r).
Features of this calculator
- Four solving modes — from any one value (r, d, C or A), from three points, from a center and a point, and from the general equation x² + y² + Dx + Ey + F = 0.
- Automatic derivation of all four quantities (radius, diameter, circumference, area) whenever any one is known, with the substituted formula shown.
- Advanced tools: arc length & sector area, chord length (from angle, chord or center-to-chord distance), circular segment area, annulus, tangent from an external point, line–circle intersection, two-circle relationship + radical line, and equivalent-square comparison.
- Live labeled SVG diagram that adapts to the current mode — showing radius, three-point circumcircle, or annulus.
- Length-unit selector (mm, cm, m, km, in, ft, yd, mi) and automatic area conversion to m², ft², yd², acres, hectares, km², mi².
- Show/hide step-by-step working for every result — formula, substitution, and final answer.
- Copy result, download as PNG or PDF, and print — all from the result toolbar.
- Input validation: collinear-point detection, chord-longer-than-diameter, non-real circles (r² ≤ 0), interior-point tangent, and more.
Frequently asked questions
+What is a circle?
A circle is the set of all points in a plane that are the same distance (the radius) from a fixed center point.
+How do I find the area of a circle from its circumference?
First find the radius from r = C / (2π), then compute area A = π r². You can also combine them directly: A = C² / (4π).
+What is π (pi)?
π is the ratio of a circle's circumference to its diameter, roughly 3.14159. It is irrational, meaning its decimal expansion never ends or repeats, and it is the same value for every circle.
+How do I find a circle from three points?
The three points are equidistant from the center. Set up two equations by equating squared distances from the unknown center to each pair of points, solve the resulting linear system for the center (h, k), then take r as the distance from (h, k) to any of the three points.
+What is the general equation of a circle?
x² + y² + Dx + Ey + F = 0 describes a circle with center (−D/2, −E/2) and radius √((D/2)² + (E/2)² − F). If that expression under the square root is 0 you get a single point; if it is negative there is no real circle.
+How long is the arc for a given central angle?
Arc length s = r · θ when θ is in radians. In degrees, s = π r θ / 180. For a full circle (θ = 2π) this reduces to the circumference 2π r.
+How is a chord length related to the central angle?
For a chord subtending a central angle θ, the chord length is 2 r sin(θ/2). Going the other way, θ = 2 · arcsin(chord / (2 r)).
+What is a circular segment?
The region between a chord and its arc. Its area equals the sector area minus the triangle formed by the two radii and the chord: A = ½ r² (θ − sin θ) with θ in radians.
+How do I compute a tangent from an external point?
If P is at distance d from the center of a circle of radius r (with d > r), the length of the tangent segment from P to the point of tangency is √(d² − r²).
+Can I express a circle's area as an equivalent square?
Yes — a square with side s = √A has the same area as a circle of area A. Constructing such a square with only compass and straightedge is impossible (the classical 'squaring the circle' problem), but the numeric equivalence is useful for comparing areas.
+How do I tell if a line and a circle intersect?
Substitute the line equation into (x − h)² + (y − k)² = r² to get a quadratic in x. Its discriminant Δ decides the outcome: Δ > 0 gives two intersection points (a secant), Δ = 0 gives one (a tangent), and Δ < 0 means the line misses the circle. The Line–Circle Intersection tool below runs this substitution and shows every step.
+How do I know if two circles intersect?
Compare the distance d between their centers to the sum and difference of their radii. If d > r₁ + r₂ the circles are separate; d = r₁ + r₂ is externally tangent; |r₁ − r₂| < d < r₁ + r₂ gives two intersection points; d = |r₁ − r₂| is internally tangent; and d < |r₁ − r₂| means one lies fully inside the other. The Two Circles tool computes the intersection coordinates and the radical line automatically.