Circle Calculator

About the Circle Calculator

Type any one of a circle’s radius, diameter, circumference or area and get the other three, with the formula and your numbers at every step. Answers stay exact when they can — a radius of 7 gives an area of 49π ≈ 153.94 — and you can work with π = 3.14 or 22/7 when a textbook asks for it.

The second tab handles parts of a circle: from any two of the radius, central angle, inscribed angle, arc length, chord, segment height (sagitta), sector area and segment area, it finds all the rest and draws the sector and segment to scale. When two different arcs fit your values — a chord and a radius fit both a minor and a major arc — it shows both. The third tab finds a circle’s equation from its centre and radius, its centre and a point, the ends of a diameter or three points, or reads an equation you type, and converts between the standard form (x − h)² + (y − k)² = r² and the general form x² + y² + Dx + Ey + F = 0.

Common uses

• Homework on circumference, area, arcs and sectors — including the π = 22/7 answers many textbooks expect.
• Finding the radius of an arch, a curved wall or a round tabletop from the width (chord) and rise (sagitta) you can measure.
• Coordinate geometry: the circle through three points, the centre and radius from an equation, the axis intercepts, and whether a point lies inside, on or outside the circle.
• Checking a sector or segment area for a pie-chart slice, a garden bed or a sheet-metal part.

Formulas used

• Circumference C = 2πr = πd. Area A = πr² = πd²/4 = C²/(4π).
• With the central angle θ in radians: arc length s = rθ, sector area K = ½r²θ, chord c = 2r·sin(θ/2), segment height h = r(1 − cos(θ/2)) and segment area G = ½r²(θ − sin θ). In degrees, s = (θ/360°) × 2πr and K = (θ/360°) × πr².
• An inscribed angle is half the central angle that stands on the same arc: α = θ/2.
• From a chord and its height: r = (c²/4 + h²)/(2h), because half the chord, r − h and r form a right triangle.

Equations of a circle

The circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². Expanding gives the general form x² + y² + Dx + Ey + F = 0 with D = −2h, E = −2k and F = h² + k² − r². Going back, the centre is (−D/2, −E/2) and r² = D²/4 + E²/4 − F. If that is negative, no real point fits (an imaginary circle); if it is 0, the “circle” is a single point. If x² and y² have the same coefficient other than 1, the tool divides by it first; if the coefficients differ, or there is an xy term, the curve is not a circle and the tool says what it is instead.

For three points, each point is put into the general form; subtracting the first equation from the other two leaves two linear equations in D and E. If the three points lie on one straight line, those equations have no solution and no circle passes through all three.

When two arcs fit

Some pairs of values fit more than one circle. A chord and a radius fit a minor arc and a major arc, with central angles θ and 360° − θ. An arc length and a segment height can fit two circles whose arcs are both longer than a semicircle. The tool scans every central angle from 0° to 360°, refines each solution it finds, and lists them all.

Limitations

• Flat (Euclidean) circles only — not circles drawn on a sphere.
• Calculations use double-precision numbers (about 15 significant digits). Exact forms such as 49π, 5√(2/π) or 6π − 9√3 are shown when your inputs are whole numbers, fractions, roots or multiples of π and the result stays short.
• In the arc tab, two values that are neither the radius nor an angle (except a chord with its height) are solved numerically, so those answers are decimals.
• With π taken as 3.14 or 22/7, an answer is only as exact as that value of π, so it is shown as a decimal (roots that do not involve π, such as a chord of 7√2, stay exact).
• Coordinates in the equation tab have no unit: the radius, area and intercepts are in the same units as the coordinates.

Privacy

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