Free · Mathematical relations and geometry
Ellipse Center, Vertices, Foci and Measures
Calculate axis-aligned ellipse vertices and foci from positive horizontal and vertical semiaxes and a center.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
With semiaxes a=5 and b=3 centered at (2, −1), the horizontal vertices are (−3, −1) and (7, −1); the foci are (−2, −1) and (6, −1).
(x−c₁)²/a²+(y−c₂)²/b²=1; focal distance=√|a²−b²|
Worked example
Enter these known values and leave the other values blank.
- a
- 5
- b
- 3
- c₁
- 2
- c₂
- -1
- Eccentricity
- 0.8
- Area
- 47.12
- Perimeter
- 25.527
- x-coordinate
- 2
- y-coordinate
- -1
- x-coordinate
- -2
- y-coordinate
- -1
- x-coordinate
- 6
- y-coordinate
- -1
- x-coordinate
- -3
- y-coordinate
- -1
- x-coordinate
- 7
- y-coordinate
- -1
- x-coordinate
- 2
- y-coordinate
- -4
- x-coordinate
- 2
- y-coordinate
- 2
Assumptions and limitations
- a is the horizontal semiaxis and b is the vertical semiaxis. The ellipse is axis-aligned; neither semiaxis must be the longer one.
- The center shifts the shape without changing its area, perimeter or eccentricity.
- The fields called vertices follow the horizontal axis and semivertices follow the vertical axis, even if the vertical axis is longer.
- Perimeter uses Ramanujan’s second approximation. Area and focal distance use exact geometric formulas.
- Coordinates and semiaxes share one length scale. The interface has no unit selectors.
Common questions
What happens if the vertical axis is longer?
The foci lie above and below the center. The named horizontal vertex fields still describe the horizontal endpoints.
References
The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.
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