Circle & Ellipse Calculator

This calculator handles two related shapes in one tool: a circle, solved from whichever single measurement you already know, and an ellipse, solved from its two semi-axes. Switch between them with the shape selector — the rest of the form adapts to match.

Circle formulas

For a circle with radius r:

Diameter      = 2r
Circumference = 2πr
Area          = πr²

You don't need to already know the radius — pick "Solve circle from" and supply whichever of radius, diameter, circumference, or area you have, and the calculator inverts the formula to find the radius first (r = d/2, r = C / 2π, or r = √(A / π)), then derives the other three from it.

If you also enter a sector angle (in degrees, up to 360), the calculator adds the arc length and sector area for that wedge of the circle:

Arc Length  = (θ / 360) × 2πr
Sector Area = (θ / 360) × πr²

Leave the angle blank to skip this — it's entirely optional.

Ellipse formulas

For an ellipse with semi-major axis a and semi-minor axis b (the calculator sorts whichever two values you enter so the larger is always treated as the major axis):

Area              = π × a × b                                  (exact)
Perimeter         ≈ π[3(a+b) − √((3a+b)(a+3b))]                 (Ramanujan's 2nd approximation)
Eccentricity      = √(1 − b²/a²)
Foci Distance (c) = √(a² − b²)
Semi-Latus Rectum = b² / a

Area has an exact formula, but an ellipse's perimeter does not — no elementary closed-form expression exists for it. This tool uses Ramanujan's second approximation, which stays accurate to within a few parts in a million across virtually any aspect ratio, making it effectively indistinguishable from the true value for practical purposes. Eccentricity measures how "stretched" the ellipse is (0 = a perfect circle, closer to 1 = more elongated); the foci are the two fixed interior points whose distances to any point on the ellipse always sum to 2a; the semi-latus rectum is the distance from a focus to the ellipse, measured perpendicular to the major axis.

How to use it

1. Choose Circle or Ellipse.
2. Pick a unit (millimeters, centimeters, meters, inches, feet, or yards) — it's a label on the results only; no unit conversion happens, so enter every value in the same unit.
3. Choose how many decimal places to display.
4. For a circle: pick which value you're solving from and enter it, plus an optional sector angle. For an ellipse: enter both semi-axes.
5. Results update instantly as you type.

FAQ

Why doesn't the ellipse have a sector calculation? Sector arc length and area only have simple formulas for a circle, where every point on the boundary is equidistant from the center. An ellipse's boundary isn't a constant distance from its center, so a sector's arc length would require numerical integration rather than a closed-form formula — outside the scope of this calculator.

How accurate is the ellipse perimeter? Ramanujan's second approximation is exact only for a true circle (a = b); for any other aspect ratio it carries a tiny relative error, typically far below 0.001%, even for very elongated ellipses. For virtually all practical, engineering, and educational purposes it's treated as exact.

Does the unit dropdown convert my measurement? No. It only labels the output — area is shown in squared units (e.g. cm²) since it measures a two-dimensional surface, while every other result stays in the plain unit you entered.

What if I enter a negative, zero, or blank value? The calculator requires a positive value for whichever measurement it's solving from (or for both ellipse axes); a blank, zero, or negative entry is flagged with a validation message asking for a value greater than 0, rather than silently returning a result.

Privacy

Everything runs locally in your browser — nothing you enter is uploaded or stored anywhere.