What This Calculator Does
You are designing an oval tabletop, sizing an elliptical window, or working through an orbital mechanics problem. This ellipse calculator takes the two semi-axes and returns the area, the perimeter (circumference), the eccentricity, and the focal distance. It automatically treats the larger value as the semi-major axis and the smaller as the semi-minor, so the order you enter them does not matter.
An ellipse is a stretched circle. Instead of one radius, it has two: the semi-major axis a (the longest radius) and the semi-minor axis b (the shortest radius). When a equals b, the ellipse is a circle. The further apart a and b get, the more elongated the shape becomes. For circle-specific calculations, see our Circle Calculator.
Inputs Required
- Semi-major axis (a): Half the longest diameter of the ellipse
- Semi-minor axis (b): Half the shortest diameter of the ellipse
Outputs Provided
- Area: The space enclosed by the ellipse
- Perimeter: The total length around the ellipse (approximate)
- Eccentricity: A measure of how stretched the ellipse is, from 0 (circle) to nearly 1 (flat)
- Focal Distance: The distance from the center to each focus
How the Calculation Works
Area: A = πab
Perimeter (Ramanujan): P ≈ π[3(a + b) − √((3a + b)(a + 3b))]
Eccentricity: e = √(1 − b²/a²)
Focal Distance: c = √(a² − b²)
The area formula is a direct extension of the circle area. A circle is πr², and an ellipse replaces one r with a and the other with b, giving πab. You can think of it as a circle of radius b that has been scaled horizontally by a factor of a/b, which stretches the area by that same factor.
The perimeter of an ellipse has no exact closed-form formula in terms of elementary functions. It requires an infinite series called a complete elliptic integral. This calculator uses Ramanujan's first approximation, which is accurate to within about 0.1% for most ellipses and gets more accurate as the shape approaches a circle. For a circle (a = b), it gives exactly 2πa. The eccentricity measures how far the ellipse deviates from a circle: 0 means a perfect circle, and values approaching 1 mean a very thin oval. The focal distance c is how far each focus sits from the center, and the two foci together define the ellipse through the property that the sum of distances from any point on the curve to the two foci equals 2a.
How to Use the Calculator
- Enter the semi-major axis, which is half the longest diameter. If you only know the full diameter, divide it by 2.
- Enter the semi-minor axis, which is half the shortest diameter.
- The calculator automatically uses the larger value as the semi-major axis, so entry order does not matter.
- Read the area, perimeter, eccentricity, and focal distance from the results panel.
Example Calculations
Example 1: An Oval Tabletop
A furniture maker builds an oval dining table with a length of 80 inches and a width of 48 inches. The semi-major axis is 40 inches and the semi-minor axis is 24 inches. The area is π × 40 × 24 = 960π ≈ 3015.93 square inches. The perimeter is approximately π[3(64) − √((120 + 24)(40 + 72))] = π[192 − √(144 × 112)] = π[192 − 126.99] ≈ 204.21 inches, which tells the maker how much edge banding to buy.
Example 2: A Planetary Orbit
Earth's orbit has a semi-major axis of about 149.6 million kilometers and a semi-minor axis of about 149.58 million kilometers. The eccentricity is √(1 − 149.58²/149.6²) ≈ 0.0167, which is very close to circular. The focal distance is √(149.6² − 149.58²) ≈ 2.5 million kilometers, meaning the Sun sits about 2.5 million kilometers off-center. This small offset is why Earth's distance from the Sun varies slightly over the year.
Real World Scenarios
Sizing an Elliptical Skylight
An architect specifies an elliptical skylight with a 60-inch by 36-inch opening. The semi-axes are 30 and 18 inches. The area is π × 30 × 18 ≈ 1696.46 square inches, which sets the glass area and the light transmission. The perimeter is approximately π[3(48) − √((108)(84))] = π[144 − 95.25] ≈ 153.21 inches, which is the length of the curved frame the fabricator must bend. Because bending an aluminum extrusion to an ellipse requires knowing the perimeter for material ordering, getting this number right avoids costly rework.
Calculating an Athletic Track Infield
A school builds a 400-meter running track with an elliptical infield. The straight sections are 84.39 meters and the curved ends have a radius of 36.5 meters, giving an infield ellipse with a semi-major axis of about 73 meters and a semi-minor axis of about 36.5 meters. The area is π × 73 × 36.5 ≈ 8365.86 square meters, which is about 2.07 acres. This tells the grounds crew how much sod to order and how much fertilizer the infield needs per season.
Designing an Elliptical Speaker Baffle
An audio engineer designs an elliptical speaker baffle with a 12-inch by 6-inch opening. The semi-axes are 6 and 3 inches. The area is π × 6 × 3 ≈ 56.55 square inches, which affects the acoustic loading. The eccentricity is √(1 − 9/36) = √0.75 ≈ 0.866, indicating a fairly elongated oval. The focal distance is √(36 − 9) ≈ 5.20 inches, which the engineer uses to position internal bracing so it avoids the foci where standing waves concentrate.
Common Mistakes to Avoid
- Entering full diameters instead of semi-axes: The formulas use the semi-major and semi-minor axes, which are half the full diameters. If you have the full length and width, divide each by 2 before entering.
- Treating the perimeter as exact: The perimeter is an approximation (Ramanujan's formula). It is accurate to about 0.1% for most ellipses, but for very elongated shapes (eccentricity above 0.9), consider a more precise method.
- Confusing eccentricity with aspect ratio: Eccentricity is √(1 − b²/a²), not b/a. A 2:1 ellipse has an aspect ratio of 0.5 but an eccentricity of about 0.866.
- Forgetting that a circle is a special ellipse: When a equals b, the area becomes πa² and the perimeter becomes 2πa, exactly matching circle formulas. The eccentricity drops to 0.
Limitations of This Calculator
This tool computes area, eccentricity, and focal distance exactly. The perimeter uses Ramanujan's first approximation, which is accurate to roughly 0.1% for ellipses with eccentricity below 0.9 and improves as the shape nears a circle. It does not compute the full elliptic integral, so for highly elongated ellipses where precision matters (such as satellite orbit analysis), use a dedicated elliptic integral calculator. The tool also does not handle rotated ellipses or partial ellipse segments.
Authoritative Research and Resources
- Wolfram MathWorld: Ellipse - A comprehensive reference on ellipse geometry, including the exact perimeter series, eccentricity, and the focal property that defines the shape.
- Maths Is Fun: Ellipse - A visual guide to ellipse properties with interactive diagrams showing how the two foci and the semi-axes relate.
For related tools, calculate circle properties with our Circle Calculator, general shape areas with our Area Calculator, or trapezoid area with our Trapezoid Calculator.