Ellipse Standard Form Calculator
Convert vertices, center coordinates, or foci to standard ellipse equation form
Calculate Ellipse Standard Form
Enter the coordinates of the four vertices (two on horizontal axis, two on vertical axis):
Horizontal Axis Vertices
Vertical Axis Vertices
Ellipse Standard Form
Standard Form Equation
Vertices (Major Axis)
V₁: (0.00, 0.00)
V₂: (0.00, 0.00)
Co-vertices (Minor Axis)
CV₁: (0.00, 0.00)
CV₂: (0.00, 0.00)
Foci
F₁: (0.00, 0.00)
F₂: (0.00, 0.00)
Orientation
Major axis: Vertical
Shape: Highly elliptical
Ellipse Visualization
Example Calculation
Example: From Vertices (±13, 0) and (0, ±12)
Horizontal vertices: V₁(-13, 0), V₂(13, 0)
Vertical vertices: V₃(0, -12), V₄(0, 12)
Center: (0, 0) - midpoint of opposite vertices
Step-by-step Solution
1. Calculate semi-major axis: a = |13 - (-13)|/2 = 13
2. Calculate semi-minor axis: b = |12 - (-12)|/2 = 12
3. Since a > b, major axis is horizontal
4. Center is at origin (0, 0)
Standard Form: x²/13² + y²/12² = 1
Standard Form Templates
Horizontal Major Axis
(x-h)²/a² + (y-k)²/b² = 1
When a > b
Vertical Major Axis
(x-h)²/b² + (y-k)²/a² = 1
When a > b
Circle (Special Case)
(x-h)² + (y-k)² = r²
When a = b
Key Relationships
Focal Distance
c² = a² - b²
Eccentricity
e = c/a = √(1 - b²/a²)
Area
A = π × a × b
Quick Tips
Center (h,k) is the midpoint of vertices
Semi-major axis (a) is always ≥ semi-minor axis (b)
Foci lie on the major axis at distance c from center
Eccentricity e = 0 for circle, 0 < e < 1 for ellipse
Sum of distances from any point to foci = 2a
Understanding Ellipse Standard Form
What is Standard Form?
The standard form of an ellipse is a mathematical equation that describes all points on the ellipse. It's written as (x-h)²/A² + (y-k)²/B² = 1, where (h,k) is the center, and A and B are related to the axis lengths.
Key Components
- •(h, k): Center coordinates of the ellipse
- •a: Semi-major axis (longest radius)
- •b: Semi-minor axis (shortest radius)
- •c: Focal distance from center to focus
Converting from Different Inputs
From Vertices
Find center as midpoint, calculate axis lengths as half the distance between opposite vertices
From Center + Axes
Directly substitute values into standard form equation
From Foci + Vertices
Calculate a from vertices, c from foci, then b = √(a² - c²)
Remember: The denominator under x² and y² determines orientation. Larger denominator indicates the direction of the major axis.