Ellipse Calculator
Enter Data:
Major Axis: x-axis
Minor Axis: y-axis
[ x1 > x2 ] Enter x1:
Enter x2:

[General Ellipse Data]
[Cartesian Data]
[Polar Data]
[Parametric Data]
[Semi-Ellipse Data]
[Section Data]
[Sector Data]
General Ellipse Data:
± y1 =
± y2 =
± Foci =
(x1, y1) to f1 =
(x1, y1) to f2 =
[(x1, y1) to Foci] Sum =
Total Area =
Total Perimeter =
Cartesian Equation Data:
x ² coefficient =
y ² coefficient =
(ab) ² =
x-intercept: a =
y-intercept: b =
Line tangent at Point (x1, y1)
Angle to x-axis =
Slope =
x-intercept =
y-intercept =
Polar Data:
Origin at Center (0,0)
Point (x1, y1)
r1 to (0,0) =
θ1 to x-axis =
Point (x2, y2)
r2 to (0,0) =
θ2 to x-axis =
Polar Data:
Eccentricity ε =
Directrix to (0,0):
Focus to (k,0):
Point (x1, y1)
r1 to focus =
θ1 to x-axis
Point (x2, y2)
r2 to focus =
θ2 to x-axis
Parametric Angles at (0,0):
Origin at Center (0,0)
Point (x1, y1)
φ1 to x-axis =
Point (x2, y2)
φ2 to x-axis =
Semi-Ellipse Data:
Scope: 1st and 2nd Quadrants
x1 = a , x2 = -a
Ix =
I =
Ax =
A =
x =
Section Data, Vertical Elements:
Scope: 1st and 2nd Quadrants
-a £ x £ a , x1 > x2
Ix =
I =
Ax =
A =
x =
Section Data, Horizontal Elements:
Scope: 1st Quadrant
0 £ x £ a , x1 > x2
Ix =
I =
Ax =
A =
x =
Section Data, Chord x1 to x2:
Scope: 1st and 2nd Quadrants
-a £ x £ a , x1 > x2
Chord =
Arc =
Ix =
I =
Ax =
[Sector - Triangle] A =
[Section - Trapezoid] A =
x =
Sector Data:
Scope: 1st and 2nd Quadrants
-a £ x £ a , x1 > x2
Triangle Area =
Ix =
I =
Ax =
[Section ± Triangles] A =
[ab(φ2 - φ1) ¸2] A =
x = [
Return to DATA ENTRY Forms]
Forms of Ellipse Equations:
Standard Form :
(x ¸ a) ² + (y ¸ b) ² = 1
Cartesian or Rectangular :
(bx) ² + (ay) ² = (ab) ²
Polar , Origin at Center :           
r = ab ¸ Ö (b cosθ) ² + (a sinθ) ²
Polar , Origin at Focus :
r = kε ¸ (1 ± ε cosθ)
Parametric Equations :
x = a cosφ , y = b sinφ

Ellipse Axes, with x-intercept a and y-intercept b

    Ellipse, showing x and y axes, semi-major axis a, and semi-minor axis b.

Sum of distances from the foci to any point is constant

    The sum of the distances for any
point P(x,y) to foci (f1,0) and (f2,0) remains constant.

Polar Equation variables: the origin is the center of the ellipse

Polar Equation: Origin at Center (0,0)

Focus-Directrix equation variables: the origin is the focus of the ellipse

Polar Equation: Origin at Focus (f1,0)

    When solving for Focus-Directrix values with this calculator, the major axis, foci and k must be located on the x-axis.
    r = kε ¸ (1 ± ε sinθ) is the equation if the major axis of the ellipse is on the y-axis. The ± sign is governed by the location of k on the x-axis.

Ellipse Sections: Integration along the x-axis with vertical elements

Integration along x-axis, Vertical elements
Scope of calculation: -a £ x £ a
First and Second Quadrants

    Centroidal axes for all section and sector calculations are measured with respect to the x-axis.

Ellipse Sections: Integration along the y-axis with horizontal elements

Integration along y-axis, Horizontal elements
Scope of calculation: 0 £ x £ a
First Quadrant

    Negative values in the Second Quadrant are actually permissible, but must be interpreted with care. The shape created about the y-axis will be asymmetrical if the entries for x and -x are not of equal magnitude.

Ellipse Sections created by chord x1-x2

Sections bounded by chord x1 to x2
Ellipse sectors minus triangular areas
Scope of calculation: -a £ x £ a
First and Second Quadrants

Ellipse Sectors

Ellipse Sectors
Scope of calculation: -a £ x £ a
First and Second Quadrants

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