Enter the semi-major and semi-minor axes — get the exact area and a highly accurate perimeter approximation.
Enter both axes to see the area and approximate perimeter
This ellipse calculator finds the area and perimeter of an ellipse from its semi-major axis (a) and semi-minor axis (b) — half the longest and shortest diameters. Unlike a circle, an ellipse has no simple exact perimeter formula, so this calculator uses Ramanujan's second approximation, one of the most accurate perimeter approximations known, alongside the exact area formula π × a × b. Whether you're checking an elliptical orbit, an architectural arch, or a geometry homework problem, this free tool computes both results instantly and clearly labels the perimeter as an approximation rather than an exact value.
Given the semi-major axis (a) and semi-minor axis (b), the calculator computes the exact area using π × a × b, then estimates the perimeter using Ramanujan's second approximation — a formula so accurate that its error is negligible for virtually all real-world ellipses.
This tool suits geometry and calculus students studying conic sections, engineers designing elliptical gears, cams, or ducting, architects working with elliptical arches or domes, astronomers estimating orbital path lengths, and anyone with an elliptical shape who needs its area or approximate perimeter quickly.
Ellipses appear throughout nature and engineering — from planetary orbits to whispering-gallery mirrors to elliptical running tracks — precisely because the shape balances directional stretching with smooth, continuous curvature. Because there's no exact perimeter formula, using a proven, highly accurate approximation like Ramanujan's avoids the larger errors that cruder formulas introduce.
Mechanical engineers compute perimeter for elliptical gear or cam profiles used in variable-speed machinery. Architects estimate material length for elliptical arches, domes, and window frames. Astronomers approximate the path length of elliptical orbits. Track and field designers plan elliptical running track dimensions. Students and teachers use it to verify ellipse geometry and conic-section homework.
Eccentricity (e) measures how far an ellipse is from being a circle — from 0 for a perfect circle up toward 1 for an extremely stretched shape. It is computed as e = √(1 − b²/a²), so the two axes you enter indirectly determine it. When a and b are far apart, the ellipse is highly eccentric, and that is exactly when the perimeter approximation's tiny error grows slightly.
Each focus sits on the major axis at distance c from the center, where c = √(a² − b²). For an ellipse with a = 5 and b = 3, the foci lie about 4 units from the center on either side. The defining property of an ellipse is that the sum of the distances from any point on the curve to the two foci is always constant, equal to 2a.
A true ellipse has a precise mathematical definition and a set of exact formulas, while "oval" is a loose everyday word for any rounded, egg-shaped or oblong outline. This calculator only applies to genuine ellipses with uniform curvature — irregular ovals, egg shapes, and other closed curves do not follow the area and perimeter formulas shown here.
Halve each diameter before entering. If the full major diameter is D and the full minor diameter is d, then a = D/2 and b = d/2, and the area is π × a × b. Entering the full diameters instead of the semi-axes is one of the most common mistakes, because this calculator expects half of each diameter.
Because h = (a − b)²/(a + b)² shrinks toward zero as the two axes converge, and at h = 0 Ramanujan's formula collapses exactly to 2πr, the true circumference of a circle. The rounder the ellipse, the closer the approximation is to perfect; the error only becomes noticeable for extremely elongated shapes.
Centered at the origin with the semi-major axis a along the x-axis and the semi-minor axis b along the y-axis, an ellipse follows (x/a)² + (y/b)² = 1. Any point (x, y) that satisfies this equation lies on the curve, and the same a and b are exactly what this calculator needs to return the area and approximate perimeter.
The exact area formula and the best-known perimeter approximation
Unlike the perimeter, an ellipse's area has a simple closed-form formula (π × a × b) with no approximation involved, just like a circle's πr².
An ellipse's perimeter requires an elliptic integral with no elementary closed form — Ramanujan's approximations are the practical, highly accurate workaround.
When the semi-major and semi-minor axes are equal, both the area and perimeter formulas collapse exactly to the familiar circle formulas, with zero approximation error.
From entering the axes to reading the area and perimeter approximation
Measure half of the ellipse's longest diameter (a).
Measure half of the ellipse's shortest diameter (b).
Type a and b into the two input fields, using consistent units.
The calculator computes the exact area (π × a × b) and estimates the perimeter using Ramanujan's second approximation.
Review both results, remembering the perimeter is a highly accurate approximation, not an exact value.
Solving an ellipse with semi-major axis 5 and semi-minor axis 3
An elliptical garden bed has a semi-major axis of 5 m and a semi-minor axis of 3 m. What are its area and approximate perimeter (for edging material)?
Explanation: The area (47.1239 m²) is exact, while the perimeter (≈25.5270 m) is Ramanujan's second approximation — accurate to within a tiny fraction of a percent of the true elliptic-integral value, more than precise enough for edging material or fencing calculations.
What each output represents and how they relate
| Property | What It Represents | Precision |
|---|---|---|
| Area | Two-dimensional space enclosed by the ellipse | Exact (π × a × b) |
| Perimeter | Total distance around the ellipse's curved edge | Highly accurate approximation (Ramanujan II) |
Reading the results together: area is an exact geometric quantity, while perimeter can only ever be approximated for a true ellipse — this calculator uses one of the most accurate known approximations, but it is still technically not exact.
Typical ranges: as a and b get closer together, the ellipse approaches a circle and the approximation error shrinks toward zero; as a grows much larger than b (a very elongated ellipse), the approximation's tiny error grows slightly but remains far smaller than simpler formulas.
Manual verification: for area, multiply a, b, and π directly. For perimeter, recompute h by hand, then carefully work through the nested square root and fraction in Ramanujan's formula step by step to reproduce the same approximate result.
Where computing ellipse area and perimeter is genuinely useful
Check ellipse area and perimeter approximation problems in geometry or calculus.
Approximate the path length of a planet's or satellite's elliptical orbit.
Compute perimeter for variable-speed elliptical gear or cam profiles.
Estimate material length for architectural elliptical arches and domes.
Plan track lap distance and lane markings for oval-shaped running tracks.
Work out elliptical reflector or acoustic mirror dimensions.
Calculate trim length needed for oval windows or decorative mirrors.
Estimate cross-sectional area and edge length for elliptical air ducts.
Compute precise oval proportions for design motifs and layouts.
Size oval-shaped product bases, labels, or packaging inserts.
Plan elliptical garden bed area and edging material needs.
Check elliptical reflector dish dimensions for signal focusing.
Estimate perimeter for oval-shaped stadium seating bowls.
What this ellipse calculator does well, and where it doesn't apply
How an ellipse compares to its special-case relative, the circle
| Shape | Axes | Area Formula | Perimeter Formula |
|---|---|---|---|
| Ellipse | Semi-major (a) ≠ semi-minor (b) | π × a × b (exact) | Ramanujan approximation (no exact form) |
| Circle | a = b = radius (r) | π × r² (exact) | 2 × π × r (exact) |
Summary: This ellipse calculator gives you an instant, free way to find the exact area and a highly accurate approximate perimeter using Ramanujan's second approximation, with every formula shown. Pair it with related tools like the Circle Calculator and Area Calculator for a fuller picture of curved-shape geometry.
Common questions about ellipse measurements
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