Ellipse Equation Calculator

Calculator Form

Choose Input Mode

The general form supports rotated ellipses with an xy term.

Center h

Center k

Semi-major axis a

Semi-minor axis b

Rotation angle θ (degrees)

Quick example

A for x²

B for xy

C for y²

D for x

E for y

F constant

Center h

Center k

Vertex x

Vertex y

Co-vertex x

Co-vertex y

Example Data Table

Case	Inputs	Key Output
Standard Example	h=0, k=0, a=6, b=4, θ=0°	Center (0,0), e≈0.745356, area≈75.398224
Rotated Example	h=1, k=-2, a=8, b=3, θ=30°	Rotated canonical equation and full general coefficients
General Form Example	0.090557x² - 0.054279xy + 0.131665y² - 1 = 0	Recovered center, axes, angle, foci, and graph
Point-Based Example	Center (1,2), Vertex (6,5), Co-vertex (-1,5.333333)	Axes are estimated from distances and directions

Formula Used

This calculator uses the rotated ellipse model, the quadratic general equation, and the parametric representation. It also derives foci, vertices, co-vertices, eccentricity, area, perimeter, and latus rectum.

1) Rotated Canonical Form

((x′)² / a²) + ((y′)² / b²) = 1
x′ = (x - h)cosθ + (y - k)sinθ
y′ = -(x - h)sinθ + (y - k)cosθ

2) General Quadratic Form

Ax² + Bxy + Cy² + Dx + Ey + F = 0

When the coefficients describe a real ellipse, the calculator finds the center, principal axes, and rotation by diagonalizing the quadratic part.

3) Ellipse Geometry

c = √(a² - b²)
e = c / a
Area = πab
Approximate Perimeter = π[3(a+b) - √((3a+b)(a+3b))]
Latus Rectum = 2b² / a

4) Parametric Form

x = h + a cos(t) cosθ - b sin(t) sinθ
y = k + a cos(t) sinθ + b sin(t) cosθ

How to Use This Calculator

Select an input mode based on your known values.
Enter either canonical values, general coefficients, or coordinate points.
Click Calculate Ellipse to solve the shape.
Read the result card shown above the form.
Review the canonical equation, general equation, and plotted graph.
Use the point table and exported report for homework, teaching, or verification.
Download CSV for data records or PDF for printable reports.

FAQs

1) What does this ellipse equation calculator solve?

It solves the ellipse center, axes, foci, vertices, co-vertices, eccentricity, area, perimeter, latus rectum, rotated canonical form, general equation, and parametric form.

2) Can it handle rotated ellipses?

Yes. You can enter a rotation angle directly or provide general quadratic coefficients with an xy term. The graph and equations update for the rotated shape.

3) What if I only know the general quadratic equation?

Use the general coefficient mode. Enter A, B, C, D, E, and F. The calculator checks whether the conic is a real ellipse and then extracts its geometry.

4) Can I use points instead of equations?

Yes. The point-based mode accepts the center, one vertex, and one co-vertex. From those points, it estimates the axes and the ellipse rotation.

5) Why does the calculator sometimes reorder a and b?

The calculator treats a as the semi-major axis and b as the semi-minor axis. If your inputs reverse them, it swaps values so the final ellipse remains mathematically consistent.

6) How is eccentricity computed?

Eccentricity uses e = c/a, where c = √(a² - b²). A larger eccentricity means a more stretched ellipse, while values closer to zero indicate a rounder shape.

7) What does the graph show?

The graph plots the solved ellipse, marks the center, and keeps equal axis scaling. This helps you verify the curve shape, orientation, and position visually.

8) What is included in the CSV and PDF downloads?

The exports include the main solved values, equation forms, coordinates, and sample parametric points. They are useful for reports, notes, assignments, and record keeping.