GEOMETRY2D GeometryMathematics Calculator
๐Ÿฅš

Ellipse

An ellipse has semi-major axis a โ‰ฅ semi-minor b. Area = ฯ€ab. Eccentricity e = โˆš(1โˆ’bยฒ/aยฒ). Planets orbit in ellipses with the Sun at one focus (Kepler).

Concept Fundamentals
A = ฯ€ab
Area
e = โˆš(1โˆ’bยฒ/aยฒ)
Eccentricity
2c = 2โˆš(aยฒโˆ’bยฒ)
Foci
Planetary orbits
Kepler

Did our AI summary help? Let us know.

Planets orbit in ellipses with the Sun at one focusโ€”Kepler's first law. Eccentricity e=0 for circle; eโ†’1 for very elongated ellipse. Area ฯ€ab generalizes circle area ฯ€rยฒ when a=b=r.

Key quantities
A = ฯ€ab
Area
Key relation
e = โˆš(1โˆ’bยฒ/aยฒ)
Eccentricity
Key relation
2c = 2โˆš(aยฒโˆ’bยฒ)
Foci
Key relation
Planetary orbits
Kepler
Key relation

Ready to run the numbers?

Why: Ellipses appear in planetary orbits (Kepler), racetracks, and architecture. Eccentricity 0 = circle; near 1 = elongated.

How: Area = ฯ€ab (extends circle ฯ€rยฒ). Perimeter โ‰ˆ ฯ€(3(a+b)โˆ’โˆš((3a+b)(a+3b))). Foci distance = 2โˆš(aยฒโˆ’bยฒ). Sum of distances to foci is constant.

Planets orbit in ellipses with the Sun at one focusโ€”Kepler's first law.Eccentricity e=0 for circle; eโ†’1 for very elongated ellipse.
Sources:Wolfram MathWorld - EllipseKhan Academy - Conic Sections

Run the calculator when you are ready.

Ellipse CalculatorEnter semi-major a and semi-minor b for area, perimeter, eccentricity
๐ŸŒ
GEOMETRY2D Ellipse

Ellipse โ€” Area, Perimeter, Eccentricity

Enter semi-major (a) and semi-minor (b). Get area, perimeter, eccentricity, and foci distance.

Circle โ†’Square โ†’

๐Ÿ“ Real-World Examples โ€” Click to Load

Calculation Settings

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

๐Ÿฅš

Ellipse area = ฯ€ab โ€” extends circle formula ฯ€rยฒ.

โ€” Formula

๐Ÿช

Kepler: planets orbit in ellipses with Sun at one focus.

โ€” Astronomy

๐Ÿ“‹ Key Takeaways

  • โ€ข An ellipse has two axes: semi-major (a) โ‰ฅ semi-minor (b)
  • โ€ข Area = ฯ€ab โ€” extends the circle formula ฯ€rยฒ
  • โ€ข Eccentricity e = โˆš(1 - bยฒ/aยฒ) โ€” 0 for circle, near 1 for elongated
  • โ€ข Foci distance = 2โˆš(aยฒ - bยฒ) โ€” sum of distances to foci is constant
  • โ€ข Planets orbit in ellipses with the Sun at one focus (Kepler)

๐Ÿ’ก Did You Know?

๐ŸŒEarth's orbit has eccentricity ~0.0167 โ€” nearly circular. Pluto's is ~0.25Source: Astronomy
๐Ÿ›๏ธWhispering galleries use ellipses โ€” sound at one focus reflects to the otherSource: Architecture
๐Ÿ”ญElliptical mirrors focus light from one focus to another in telescopesSource: Optics
๐ŸˆAmerican football cross-section is an ellipse โ€” aโ‰ˆ7in, bโ‰ˆ4.25inSource: Sports
๐ŸชHalley's Comet has eโ‰ˆ0.967 โ€” one of the most elongated orbitsSource: Orbital Mechanics
๐Ÿ“A circle is an ellipse with a=b, so e=0 and both foci coincideSource: Geometry

๐Ÿ“– Formulas Explained

Area

A = ฯ€ab extends the circle area ฯ€rยฒ when a = b = r.

A=ฯ€abA = \pi a b

Perimeter (Ramanujan)

No closed form; Ramanujan's approximation is accurate to <0.04%.

Pโ‰ˆฯ€(3(a+b)โˆ’(3a+b)(a+3b))P \approx \pi\left(3(a+b) - \sqrt{(3a+b)(a+3b)}\right)

Eccentricity & Foci

e measures elongation; c = โˆš(aยฒ - bยฒ) is distance from center to each focus.

e=1โˆ’b2a2,c=a2โˆ’b2e = \sqrt{1 - \frac{b^2}{a^2}}, \quad c = \sqrt{a^2 - b^2}

๐ŸŽฏ Expert Tips

๐Ÿ’ก Unit Consistency

Keep a and b in the same units. Area will be in unitยฒ.

๐Ÿ’ก Circle Check

When a = b, you get a circle: e = 0, foci at center.

๐Ÿ’ก Orbit Eccentricity

Earth ~0.017, Mars ~0.09, Mercury ~0.21. Higher = more elongated.

๐Ÿ’ก Arch Design

Elliptical arches distribute load efficiently โ€” used in bridges and domes.

โš–๏ธ Comparison Table

FeatureThis CalculatorBasicManual
Area, perimeter, eccentricity, fociโœ…โš ๏ธ LimitedโŒ
Step-by-step solutionsโœ…โŒโœ…
Interactive chartsโœ…โŒโŒ
Copy & shareโœ…โŒโŒ
7 real-world examplesโœ…โŒโŒ

๐Ÿ“Š Ellipse Quick Facts

ฯ€ab
Area Formula
0โ€“1
Eccentricity
2
Foci
a โ‰ฅ b
Axis Rule

โ“ FAQ

What is the difference between ellipse and oval?

An ellipse is a precise conic section. An oval is any egg-like shape; not all ovals are ellipses.

When is an ellipse a circle?

When a = b. Then e = 0, and both foci are at the center.

Why do planets orbit in ellipses?

Kepler's first law: orbits are ellipses with the Sun at one focus. Newton's gravity explains why.

How accurate is the perimeter formula?

Ramanujan's approximation has error <0.04%. Exact perimeter requires elliptic integrals.

What is the whispering gallery effect?

In elliptical rooms, sound at one focus reflects to the other โ€” e.g., St. Paul's Cathedral.

Can the semi-minor exceed semi-major?

No. By convention a โ‰ฅ b. Swap them if your measurements are reversed.

What are foci used for?

Reflective property: light/sound from one focus reflects to the other. Used in telescopes and acoustics.

How do I find b from area and a?

b = A / (ฯ€a). Rearrange A = ฯ€ab.

โš ๏ธ Disclaimer: Results use standard Euclidean geometry. Perimeter is an approximation. Verify for engineering applications.

๐Ÿ‘ˆ START HERE
โฌ…๏ธJump in and explore the concept!
AI

Related Calculators

Hexagon Calculator

Calculate area, perimeter, and other properties of regular hexagons with our free online calculator.

2D Geometry

Golden Rectangle Calculator

Calculate dimensions, area, and other properties of golden rectangles based on the golden ratio (ฯ† โ‰ˆ 1.618).

2D Geometry

Octagon Calculator

Octagon Calculator - Calculate area, perimeter, and other properties of an octagon with step-by-step solutions

Mathematics

Quadrilateral Calculator

Calculate area, perimeter and other properties of quadrilaterals including squares, rectangles, rhombuses, parallelograms, trapezoids, kites, and general...

Mathematics

Cube Calculator

Cube Calculator - Calculate volume, surface area, and other properties of a cube with step-by-step solutions

Mathematics

Cylinder Calculator

Cylinder Calculator - Calculate volume, surface area, and other properties of a right cylinder with step-by-step solutions

Mathematics