Goal
Find the focal distance, horizontal foci, and eccentricity from an ellipse center, semimajor axis, and semiminor axis.
Foci of an Ellipse
Find the focal distance, horizontal foci, and eccentricity from an ellipse center, semimajor axis, and semiminor axis.
Key facts
- What it does
- Find the focal distance, horizontal foci, and eccentricity from an ellipse center, semimajor axis, and semiminor axis.
- Formula
c=sqrt(a^2-b^2), foci=(h-c,k) and (h+c,k), eccentricity e=c/a, with a>=b>0.- You enter
- Center h · Center k · Semimajor axis a · Semiminor axis b
- Worked example
- c=sqrt(25-16)=3, so the foci are (-1,3) and (5,3), focus separation is 6, and eccentricity is 0.6.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Center h · Center k · Semimajor axis a · Semiminor axis b
Method
c=sqrt(a^2-b^2), foci=(h-c,k) and (h+c,k), eccentricity e=c/a, with a>=b>0.
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Foci of an Ellipse
Find the focal distance, horizontal foci, and eccentricity from an ellipse center, semimajor axis, and semiminor axis.
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Result
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Calculation map
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01
Inputs (4)
- Center h Ready
- Center k Ready
- Semimajor axis a Ready
- Semiminor axis b Ready
02
Formula
c=sqrt(a^2-b^2), foci=(h-c,k) and (h+c,k), eccentricity e=c/a, with a>=b>0.
Bounded, transparent calculation
03
Result
- Calculate to preview the result.
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Formula, assumptions, and example
Formula: c=sqrt(a^2-b^2), foci=(h-c,k) and (h+c,k), eccentricity e=c/a, with a>=b>0.
For the horizontal-major ellipse convention, c is the center-to-focus distance. The two foci lie symmetrically on the horizontal major axis, their separation is 2c, and eccentricity is the ratio c/a.
- The ellipse is axis-aligned with a horizontal semimajor axis and a vertical semiminor axis.
- a and b are finite positive lengths satisfying a >= b, so c is real and 0 <= e < 1.
- The foci describe the ideal ellipse; no rotated shape, fitted data, uncertainty, or directrix calculation is included.
Worked example: c=sqrt(25-16)=3, so the foci are (-1,3) and (5,3), focus separation is 6, and eccentricity is 0.6.
Displayed input contract
- Center h · minimum -1000000 · maximum 1000000
- Center k · minimum -1000000 · maximum 1000000
- Semimajor axis a · minimum 1.0E-6 · maximum 1000000
- Semiminor axis b · minimum 1.0E-6 · maximum 1000000
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Foci of an Ellipse for a real question
Find the focal distance, horizontal foci, and eccentricity from an ellipse center, semimajor axis, and semiminor axis. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes ellipse foci, focal distance, ellipse eccentricity. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Center h · Center k · Semimajor axis a · Semiminor axis b. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. The ellipse is axis-aligned with a horizontal semimajor axis and a vertical semiminor axis.
Need a wider view? Browse Math Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Foci of an Ellipse
- Enter Center h — The x-coordinate of the ellipse center. (coordinate units).
- Enter Center k — The y-coordinate of the ellipse center. (coordinate units).
- Enter Semimajor axis a — Positive horizontal semimajor axis length. (length units).
- Enter Semiminor axis b — Positive vertical semiminor axis length no greater than a. (length units).
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
c=sqrt(a^2-b^2), foci=(h-c,k) and (h+c,k), eccentricity e=c/a, with a>=b>0.
For the horizontal-major ellipse convention, c is the center-to-focus distance. The two foci lie symmetrically on the horizontal major axis, their separation is 2c, and eccentricity is the ratio c/a.
Worked example
c=sqrt(25-16)=3, so the foci are (-1,3) and (5,3), focus separation is 6, and eccentricity is 0.6.
Assumptions and limits
- The ellipse is axis-aligned with a horizontal semimajor axis and a vertical semiminor axis.
- a and b are finite positive lengths satisfying a >= b, so c is real and 0 <= e < 1.
- The foci describe the ideal ellipse; no rotated shape, fitted data, uncertainty, or directrix calculation is included.
Who uses this calculator?
- Students studying conic sections and ellipse definitions
- Tutors checking focal geometry and eccentricity
- Developers needing explicit focus coordinates for a horizontal ellipse
When is it useful?
- Compute both foci from center and two axis lengths.
- Compare center-to-focus and focus-to-focus distances.
- Check the circle limit where equal axes make the foci coincide.
Context and background
The mathematical structure behind the tools
Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.
Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
The foci of an ellipse are two special points that encode its focal geometry. This calculator uses a horizontal-major ellipse convention: h and k locate the center, a is the semimajor axis, and b is the semiminor axis. It computes the center-to-focus distance c=sqrt(a^2-b^2), places the two foci at (h-c, k) and (h+c, k), reports their separation, and calculates eccentricity e=c/a. The input rule a>=b>0 is enforced because it keeps c real and preserves the declared axis meaning. The guide explains the geometry, worked values, circle limit, validation, output meaning, source boundary, assumptions, and conservative limits.
What the foci result represents
An ellipse can be characterized by two fixed focal points and a constant sum of distances to them. This page does not ask for a test point or a distance sum. Instead, it derives the two focal coordinates from the center and semiaxes of an ideal horizontal ellipse. The primary distance c is measured from the center to either focus. Because the foci are symmetric, their focus-to-focus separation is 2c, which the output reports separately so the word focal distance is not left ambiguous.
Eccentricity e=c/a is dimensionless and describes how far the foci move relative to the semimajor axis. A value of zero is the circle limit, while values closer to one indicate a more elongated ideal ellipse within this finite model. The number is geometric; it is not a quality score, a likelihood, or evidence that a measured object follows an exact conic.
- c is center-to-focus distance.
- 2c is the separation between the two foci.
- Foci are placed symmetrically on the horizontal major axis.
- Eccentricity is dimensionless and equals c/a.
Read the center and axis fields
The center fields h and k form (h, k). The semimajor field a is the horizontal distance from the center to the endpoint of the major axis, and b is the vertical semiminor distance. This is a directional convention, not merely a notation preference. If a is entered smaller than b, the handler rejects the input rather than rotating the ellipse or swapping the fields. Automatic swapping would change the meaning of the visitor's values and could make a later focus placement silently answer a different problem.
All fields are finite and bounded. h and k may be negative, zero, or positive. a and b must be strictly positive, and b cannot exceed a. Units are generic length units and should be compatible across center coordinates and axes. The pure handler does not accept numeric strings or infer a unit conversion. A value just outside a displayed bound is rejected even if the same geometry would be mathematically meaningful at a larger scale.
- The center is (h, k).
- a is horizontal semimajor length.
- b is vertical semiminor length and cannot exceed a.
- Unit and coordinate-frame preparation happens before entry.
Deriving c from the axis lengths
For a horizontal-major ellipse, the center, one focus, and the endpoint of the minor axis form a right-triangle relationship in the standard construction. The semimajor length is a, the semiminor length is b, and the center-to-focus length is c. Therefore a^2=b^2+c^2 and c=sqrt(a^2-b^2). The handler calculates the difference, clamps only the exact mathematical equality through max with zero, and then checks the square root for finiteness. The a>=b validation occurs first.
Once c is known, subtract it from h for the left focus and add it to h for the right focus. Both y-coordinates remain k because the convention places the foci horizontally. The eccentricity divides c by a, so a positive denominator is guaranteed by the input rule. The output retains enough separate components for a reviewer to see that the foci were not accidentally placed on the vertical axis.
- The relation is a^2=b^2+c^2.
- c is the nonnegative square root of a^2-b^2.
- Focus x values are h-c and h+c.
- Both focus y values equal k.
A worked focal example
Use center (2,3), a=5, and b=4. The squared axis difference is 25-16=9, so c=3. The left focus is (2-3,3)=(-1,3), and the right focus is (2+3,3)=(5,3). Their separation is 6. Eccentricity is 3/5=0.6. The foci are inside the ellipse because c is smaller than a, and their midpoint is the center: ((-1+5)/2, (3+3)/2)=(2,3).
The constant-sum property offers a conceptual check. At the right horizontal endpoint (7,3), the distances to the foci are 8 and 2, which sum to 10=2a. At the top endpoint (2,7), each distance is 5, again summing to 10. The calculator does not accept a test point or calculate those distances, but preserving the example makes the returned focus coordinates independently understandable.
- The axis difference is 9 and c is 3.
- The foci are (-1,3) and (5,3).
- Focus separation is 6.
- Endpoint distance sums equal 2a in the ideal example.
The equal-axis circle boundary
When a=b, a^2-b^2 is zero, so c=0. Both focus coordinates become (h, k), and eccentricity is zero. This is the circle limit: the two focal points coincide at the center because a circle has no distinguished pair of separated foci. The input is valid because a and b are still positive and satisfy a>=b. Returning coincident foci is more informative than rejecting the case or inventing a tiny separation.
The circle limit also checks the numeric branches. Focus-to-focus distance should be exactly zero and should not appear as negative zero. Eccentricity should be finite zero. A small difference between a and b creates a small positive c; whether that difference is meaningful in measured data is outside the handler. The pure result describes the exact values supplied, not an uncertainty interval around them.
- Equal positive axes produce c=0.
- Both foci then equal the center.
- Eccentricity is zero in the circle limit.
- Near-equality sensitivity is not an uncertainty estimate.
Invalid domains and finite outputs
A negative or zero semimajor axis is rejected by the positive field contract. A negative or zero semiminor axis is rejected for the same reason. If b>a, the handler raises a model-specific error rather than taking a square root of a negative number or silently swapping the axes. These failures are different from an ordinary numeric-bound failure: they say the values are numbers but do not describe the declared horizontal-major ellipse.
Center translations can place a focus farther from zero than the center coordinate, so the handler checks focus x values after deriving them. It also checks c, 2c, and e for finiteness. Results are numeric entries with descriptive labels and no negative zero. The renderer can show the two foci as separate coordinate components without parsing a coordinate string, while the steps provide the compact geometric summary.
- Require a>=b>0 before taking the square root.
- Do not rotate or swap invalid axes silently.
- Derived focus coordinates are finite-checked.
- The output labels distinguish c, 2c, focus coordinates, and e.
Orientation, related ellipse tools, and checks
This page chooses a horizontal major axis. A vertical-major ellipse can be handled by a separate contract that places foci at (h, k-c) and (h, k+c), but this page does not infer that orientation from an alternate field order. The standard-form ellipse page preserves horizontal and vertical semiaxes without requiring a>=b, while the ellipse circumference page estimates a boundary length. Foci, equation, and circumference are related but not interchangeable outputs.
Useful checks include midpoint symmetry, focus separation equal to 2c, eccentricity between zero and one, and the identity c^2+b^2=a^2. Swapping the input point labels is not relevant because the two foci are returned in a fixed left-to-right order. If a result appears outside the ellipse, inspect whether the center and axis units share one scale before blaming the formula.
- The major axis is horizontal by contract.
- A vertical-major variant needs different focus coordinates.
- Check midpoint, separation, eccentricity, and Pythagorean identity.
- Do not compare focus results across incompatible unit scales.
Source boundary and model assumptions
The ellipse reference associated with this record is private catalog metadata used to review the standard focal relationship. The article and examples are original WorldCalculate content. The source confirms the mathematical definitions but does not validate a visitor's observed ellipse, coordinate extraction, or data precision. No outside page body, code, branding, defaults, or calculator data is used. Formula provenance therefore remains narrower than application validation.
The model assumes an ideal flat Euclidean ellipse with a horizontal semimajor axis, positive finite lengths, and an exact center. It does not model rotation, a noisy conic fit, a three-dimensional projection, or uncertainty. Eccentricity is not a diagnostic or a performance measure. Use qualified domain review when the foci feed a physical design, orbit interpretation, imaging workflow, or another high-consequence conclusion.
- Private source metadata is a formula-review trace.
- The ellipse is ideal, Euclidean, and horizontal-major.
- Fitting, rotation, and uncertainty are excluded.
- Eccentricity does not establish a physical conclusion.
Frequently asked questions and conservative limits
What does c mean here? It is the distance from the center to either focus. The distance between the foci is 2c, and both are reported so the convention is explicit. Why must a be at least b? The formula and focus placement assume a is the horizontal semimajor length; reversing the order would make the declared square-root expression non-real or require a different orientation. Why is e never negative? c is chosen as the nonnegative square root and a is positive.
Can the result identify an ellipse from only two axis lengths? It describes the ideal ellipse determined by the supplied center and axes, but it does not establish that a real dataset follows that shape. Can I use the foci for a vertical ellipse? Not with this page's horizontal convention. The conservative limit is an exact focus construction for a bounded, axis-aligned, horizontal-major model, not a general conic-fitting or physical inference tool.
- c is center-to-focus distance, while 2c is focus separation.
- a>=b preserves the horizontal-major convention.
- The result describes an ideal ellipse, not a fitted dataset.
- Vertical orientation requires a separately declared model.
The constant-sum focus property
For any point on an ideal ellipse, the sum of its distances to the two foci is 2a. The calculator does not ask for a test point, but this property explains why the focus separation and semimajor axis belong together. With foci at (h-c, k) and (h+c, k), their midpoint is the center and their total separation is 2c. The farther the foci move from the center, the more elongated the ellipse becomes while the same constant-sum rule remains in force.
At a horizontal endpoint, one focus is closer and the other is farther by amounts that add to 2a. At a point on the perpendicular minor-axis endpoint, the two distances are equal by symmetry. These are useful conceptual checks of the returned coordinates. They do not convert the page into a point-distance calculator and should be evaluated with a declared tolerance when decimal coordinates are involved.
- Boundary points have focus-distance sum 2a.
- The foci midpoint is the ellipse center.
- Focus separation is 2c.
- Symmetry gives equal distances on the minor-axis line.
Eccentricity as a shape ratio
Eccentricity is e=c/a. Because c is nonnegative and a is positive, the regular horizontal ellipse has 0<=e<1 under finite positive a and b. The circle limit gives e=0. As b becomes smaller relative to a, c approaches a and e approaches one without reaching it for positive b. This ratio describes the ideal shape's focal displacement relative to its semimajor scale; it does not describe measurement quality, orbital risk, or a probability.
The ratio has no length unit. Scaling h, k, a, and b by a common length factor scales c and a together, leaving e unchanged. Translating the center changes focus coordinates but does not change c or e. These invariants are useful when comparing similar ellipses. If a and b came from a noisy fit, uncertainty in e should be assessed separately rather than inferred from the number of displayed decimals.
- e is dimensionless.
- The circle has e=0.
- Positive b keeps e below one.
- Translation changes locations, not eccentricity.
Focus coordinates and axis orientation
The horizontal-major convention fixes the focus y-coordinate at k and changes only x by minus or plus c. Thus the left focus is (h-c, k) and the right is (h+c, k). A negative center or a focus coordinate crossing zero does not change the orientation. The output labels each coordinate separately so a renderer does not have to parse a tuple or guess which focus is left. Focus 1 is always the lower-x focus under this convention.
A vertical-major ellipse would place the foci at (h, k-c) and (h, k+c). That is not a hidden branch in this handler. The input and article explicitly choose horizontal a and vertical b with a>=b. If a user has vertical-major data, swapping fields or interpreting the returned x locations as vertical foci would be a contract error. Use a separately reviewed vertical-orientation calculator if needed.
- Both focus y-coordinates equal k.
- Focus 1 has x=h-c.
- Focus 2 has x=h+c.
- Vertical-major placement is outside this handler.
Scaling and translation checks
If the ellipse is translated by changing h or k while a and b stay fixed, c and eccentricity stay fixed and both focus coordinates translate by the same amount. This is a direct check that the center fields describe position rather than shape. If all lengths are scaled by q, then c scales by q, focus separation scales by q, and e remains unchanged. A result that violates these relationships suggests a field mix-up or a unit mismatch.
The relation c^2+b^2=a^2 is another invariant of the horizontal-major construction. Compute c from the squared difference, square it, and compare c^2+b^2 with a^2 within a tolerance. Exact equality may be affected by floating-point arithmetic for decimals. This check is more informative than comparing a formatted focus coordinate alone because it tests the relationship that defines the focal distance.
- Translation moves the foci with the center.
- Uniform length scaling changes c but not e.
- Focus separation scales with the ellipse.
- The identity c^2+b^2=a^2 checks the construction.
Focus results are not all ellipse properties
Knowing the foci does not by itself return the ellipse equation, circumference, or area. The semimajor and semiminor axes remain needed to describe the full standard form and area. Circumference is an approximation problem with its own formula. The current output includes enough focal values to understand the conic relationship, but it does not silently add those neighboring measurements. Separate records keep the visitor's requested output and validation rules clear.
The foci also do not identify a physical force center, sensor location, or orbital focus without an application model. In some contexts an ellipse is a path, but the geometry alone does not provide time, speed, mass, or a governing law. The note and assumptions limit the result to an ideal axis-aligned ellipse. Preserve that context when sharing focus coordinates with a downstream calculation.
- Foci do not replace semiaxes or the equation.
- Circumference and area are separate outputs.
- A focus is not automatically a physical force center.
- Application interpretation needs a declared model.
Focused review and edge tests
A focused test should use a=5 and b=4 to verify c=3, foci (-1,3) and (5,3), separation 6, and eccentricity 0.6. It should use equal axes to verify coincident foci and zero eccentricity. It should reject b>a with the exact horizontal-major domain error, along with zero, negative, nonfinite, and out-of-range values. These tests cover the square-root, translation, equality, and model-specific validation branches.
The generic robustness sweep should invoke the default and bounded values and assert that every numeric result is finite and not negative zero. The catalog test should verify field order h, k, a, b and the example keys. The registry test then confirms that `foci-of-ellipse` reaches this handler once, without being shadowed by the standard-form or circumference maps. Testing the registry matters because similar titles and shared symbols can otherwise hide a wrong route binding.
- Known 5-4 axes give c=3.
- Equal axes exercise the circle boundary.
- b>a must be rejected.
- Focused and registry tests should both cover the exact ID.
Foci as an ellipse review invariant
The returned values can be reviewed without redrawing the ellipse. The two focus x-coordinates should be equally distant from h, their y-coordinates should both equal k, and the midpoint should return (h, k). Their separation should equal 2c, while the focal distance should satisfy c^2=a^2-b^2. These relationships test the complete result shape and make a swapped center coordinate or misplaced vertical focus visible even when the individual numbers look finite.
The output also keeps length and dimensionless results distinct. c, 2c, and focus coordinates use length units, while eccentricity has no unit. A renderer should not append a length label to e or treat focus separation as eccentricity. When sharing the result, preserve the horizontal-major assumption and the original axis values. A mathematically consistent focus tuple still describes the wrong object if the input frame or axis orientation was misidentified.
- Focus midpoint should equal the center.
- Focus separation should equal 2c.
- c^2+b^2 should equal a^2 within tolerance.
- Length and dimensionless outputs need distinct labels.
Preserving the focal convention in a handoff
A stored focus result should include the center, a, b, the horizontal-major convention, and the returned c values. Without those fields, a left and right coordinate can be mistaken for a vertical focus pair or an unrelated point pair. The result labels already separate center-to-focus distance, focus separation, coordinates, and eccentricity, but an exported record should preserve those meanings rather than reducing everything to an unlabeled number list.
The foci are useful for explaining an ideal conic, yet they do not validate how an ellipse was observed. A later application may need a fitted orientation, measurement covariance, or a physical law that uses the foci. Those additions should be explicit. The pure handler remains a bounded construction: it derives two horizontal points and a dimensionless ratio from four finite inputs, then leaves interpretation and uncertainty to the surrounding workflow.
- Store center and both axis values with the foci.
- Preserve horizontal-major orientation in exported data.
- Keep length labels separate from eccentricity.
- Fitting and physical interpretation need separate models.
Frequently asked questions
What is the Foci of an Ellipse?
Find the focal distance, horizontal foci, and eccentricity from an ellipse center, semimajor axis, and semiminor axis.
What is the formula for the Foci of an Ellipse?
c=sqrt(a^2-b^2), foci=(h-c,k) and (h+c,k), eccentricity e=c/a, with a>=b>0. For the horizontal-major ellipse convention, c is the center-to-focus distance. The two foci lie symmetrically on the horizontal major axis, their separation is 2c, and eccentricity is the ratio c/a.
What do I need to use this calculator?
Enter Center h, Center k, Semimajor axis a, Semiminor axis b, then choose Calculate.
What are the limits of this calculator?
The ellipse is axis-aligned with a horizontal semimajor axis and a vertical semiminor axis. a and b are finite positive lengths satisfying a >= b, so c is real and 0 <= e < 1. The foci describe the ideal ellipse; no rotated shape, fitted data, uncertainty, or directrix calculation is included.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
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