Calculator Inputs
Formula Used
Major axis length: 2a
Minor axis length: 2b
Focal distance: c = √(a² - b²)
Eccentricity: e = c / a
Area: πab
Latus rectum: 2b² / a
Ramanujan perimeter estimate: π(a+b)[1 + 3h/(10 + √(4 - 3h))], where h = (a-b)²/(a+b)².
How to Use This Calculator
- Enter the ellipse center as
handk. - Select whether the major axis is horizontal or vertical.
- Enter the semi-major axis
a. - Enter the semi-minor axis
b. - Keep
agreater than or equal tob. - Add a unit label if needed.
- Choose decimal places for rounded output.
- Press Calculate to show results above the form.
- Use CSV or PDF export for reports.
Example Data Table
| Center | Orientation | a | b | c | Foci | Major Axis |
|---|---|---|---|---|---|---|
| (0, 0) | Horizontal | 5 | 3 | 4 | (-4, 0), (4, 0) | 10 |
| (2, -1) | Vertical | 6 | 4 | 4.4721 | (2, -5.4721), (2, 3.4721) | 12 |
| (3, 2) | Horizontal | 8 | 5 | 6.245 | (-3.245, 2), (9.245, 2) | 16 |
Understanding Foci and Major Axis
An ellipse looks simple, yet it carries rich geometric meaning. Its major axis is the longest line through the center. The two endpoints of this line are called vertices. The foci sit inside the ellipse on the same axis. Every point on the ellipse keeps a constant total distance from both foci. That constant total equals the full major axis length.
Why These Values Matter
Foci and axis values appear in analytic geometry, orbital modeling, optics, architecture, and computer graphics. A designer may use them to shape arches. A student may use them to confirm an equation. A physics learner may compare an orbit with an ideal ellipse. Clear values help reduce mistakes when formulas look similar.
Horizontal and Vertical Ellipses
The calculator supports both main orientations. For a horizontal ellipse, the major axis runs left and right. The foci and vertices change the x coordinate while y remains fixed. For a vertical ellipse, the major axis runs up and down. The foci and vertices change the y coordinate while x remains fixed.
Interpreting the Output
The semi major value is a. The semi minor value is b. The focal distance is c. These values satisfy c² = a² - b². Eccentricity is c divided by a. A low eccentricity means the ellipse is close to a circle. A high eccentricity means the ellipse is stretched. The result also shows standard equation form, focal coordinates, vertices, co-vertices, latus rectum, area, and perimeter estimate.
Practical Accuracy Notes
Use the same units for all length inputs. The calculator accepts decimals and negative center coordinates. The semi major axis must be greater than or equal to the semi minor axis. When both are equal, the shape becomes a circle. In that case, both foci meet at the center and eccentricity becomes zero.
Common Study Uses
This tool is useful when a problem gives center form, axis lengths, or a diagram. It can confirm the missing foci before graphing. It can also prepare neat values for worksheets. The step panel explains each substitution, so learners can trace the calculation instead of copying a final answer. This makes review faster and cleaner.
FAQs
What is the major axis of an ellipse?
The major axis is the longest diameter of an ellipse. It passes through the center, both vertices, and both foci. Its length is equal to two times the semi-major axis.
What are foci?
Foci are two fixed points inside an ellipse. For every point on the ellipse, the sum of distances to both foci stays constant.
How is focal distance calculated?
Focal distance is calculated with c = √(a² - b²). Here, a is the semi-major axis and b is the semi-minor axis.
Can the major axis be vertical?
Yes. If the ellipse is taller than it is wide, the major axis is vertical. The foci then move above and below the center.
What happens when a equals b?
The ellipse becomes a circle. The focal distance becomes zero. Both foci meet at the center, and eccentricity is also zero.
What is eccentricity?
Eccentricity measures how stretched an ellipse is. It equals c divided by a. Values near zero look rounder. Values closer to one look longer.
Why must a be greater than b?
The value a represents the semi-major axis, so it must be the larger radius. If b is larger, swap the values or change the orientation.
Does this calculator show the standard equation?
Yes. It builds the standard ellipse equation from the center, axis lengths, and orientation. It also lists foci, vertices, and related measures.