Ellipse Calculator
Calculate ellipse properties including area, perimeter, eccentricity, and focal distance
Calculate Ellipse Properties
Longest radius of the ellipse
Shortest radius of the ellipse
Ellipse Properties
Key Formulas
Area: A = πab
Perimeter: P ≈ π(a+b)(1+3h/(10+√(4-3h)))
Eccentricity: e = c/a
Focal distance: c = √(a²-b²)
where h = ((a-b)/(a+b))²
Standard Equation
x²/a² + y²/b² = 1
where a > b > 0
Center at origin (0,0)
Foci at (±c, 0)
Example Calculation
Garden Ellipse Design
Problem: Design an elliptical garden bed
Given: Semi-major axis a = 8 m, Semi-minor axis b = 5 m
Find: Area and perimeter for material planning
Solution Steps
1. Area: A = πab = π × 8 × 5 = 40π ≈ 125.66 m²
2. Calculate h: h = ((a-b)/(a+b))² = ((8-5)/(8+5))² = (3/13)² ≈ 0.0533
3. Perimeter: P ≈ π(a+b)(1+3h/(10+√(4-3h)))
4. P ≈ π(13)(1+3(0.0533)/(10+√(4-3(0.0533)))) ≈ 41.84 m
Additional Properties
Focal distance: c = √(a²-b²) = √(64-25) = √39 ≈ 6.24 m
Eccentricity: e = c/a = 6.24/8 ≈ 0.78
Foci locations: (±6.24, 0) meters from center
Ellipse Components
Semi-major Axis
Longest radius from center
Half the major axis length
Semi-minor Axis
Shortest radius from center
Half the minor axis length
Focal Distance
Distance from center to focus
c = √(a²-b²)
Eccentricity
Measure of ellipse "flatness"
e = c/a (0 < e < 1)
Eccentricity Guide
e = 0
Perfect circle
0 < e < 0.5
Nearly circular ellipse
0.5 ≤ e < 0.8
Moderately flattened
0.8 ≤ e < 1
Highly flattened
e = 1
Parabola (degenerate case)
Understanding Ellipses
What is an Ellipse?
An ellipse is a closed curve that is the locus of all points where the sum of distances to two fixed points (called foci) is constant. It's essentially a "stretched circle" and is one of the fundamental conic sections in mathematics.
Key Properties
- •Two foci (F₁ and F₂) define the ellipse
- •Sum of distances from any point to both foci is constant
- •Has two axes of symmetry
- •Eccentricity determines the shape (0 < e < 1)
Mathematical Relationships
Standard Form
x²/a² + y²/b² = 1
Fundamental Relation
c² = a² - b²
Area Formula
A = πab
Applications: Planetary orbits, architectural design, engineering (gears, optics), satellite dishes, and many natural phenomena follow elliptical patterns.