Ellipse Calculator - Axes, Foci, Area

Calculate ellipse properties including axes lengths, foci positions, and area. Essential for geometry students, engineers, and designers working with elliptical shapes.

Semi-major axis (a)

Semi-minor axis (b)

Results

Major axis: -

Minor axis: -

Foci distance (c): -

Area: -

Eccentricity (e): -

About Ellipses

An ellipse is a closed curve surrounding two focal points where the sum of distances from any point on the curve to the two foci is constant. Ellipses have important applications in astronomy, physics, engineering, and architecture.

Calculation Formulas

Major axis = 2a (longest diameter)
Minor axis = 2b (shortest diameter)
Foci distance (c) = √(a² - b²)
Area = πab
Eccentricity (e) = c/a (0 ≤ e < 1)

Example Calculation

For an ellipse with semi-major axis a=5 and semi-minor axis b=3:

Major axis = 2 × 5 = 10 units
Minor axis = 2 × 3 = 6 units
Foci distance = √(5² - 3²) = √(25-9) = √16 = 4 units
Area = π × 5 × 3 ≈ 47.1239 square units
Eccentricity = 4/5 = 0.8

Interactive Exploration

Adjust eccentricity (fixed a=5):

0 (Circle) 0.6 0.99 (Highly elliptical)

Frequently Asked Questions

What if a and b are equal?

When a = b, the ellipse becomes a circle with eccentricity 0. The foci coincide at the center.

Can b be larger than a?

By convention, a always represents the semi-major axis (larger value) and b the semi-minor axis (smaller value). If you enter b > a, the calculator will automatically swap them.

What are practical applications of ellipses?

Ellipses are used in planetary orbits (Kepler's laws), whispering galleries, satellite dishes, architechtural designs, and machine parts like elliptical gears.

Disclaimer

For informational and educational purposes only. Results may vary in practical applications due to measurement precision and other factors. Not for critical engineering or scientific use without verification.

References:

Conic Sections, Euclidean Geometry, Kepler's Laws of Planetary Motion, Architectural Geometry Principles

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Ellipse Calculator - Axes Foci Area

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