# What are the ellipse’s area and foci?

Finds an ellipse’s area, perimeter (circumference), foci, vertices, eccentricity and standard-form equation from its two semi-axes and its center.

- Page: https://www.acalculator.org/math/ellipse-calculator
- JSON spec: https://www.acalculator.org/math/ellipse-calculator.json
- Version: 59bd8589f30e

## Default answer

Example with the default inputs (Semi-axis along x (a) 3, Semi-axis along y (b) 5, Center x (h) 0, Center y (k) 0): An ellipse with semi-axes 3 and 5 has an area of 47.1238898 and a perimeter of 25.52699886.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | Semi-axis along x (a) | Half the width of the ellipse: the distance from the center to its edge along x. |
| b | Semi-axis along y (b) | Half the height of the ellipse: the distance from the center to its edge along y. |
| h | Center x (h) | The x coordinate of the center; 0 when left empty. |
| k | Center y (k) | The y coordinate of the center; 0 when left empty. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| area | Area | The space inside the ellipse, πab. |
| perimeter | Perimeter (circumference) | The distance around the ellipse, exact to double precision. |
| equation | Equation | The standard form of the ellipse. |
| foci | Foci | The two focus points on the longer axis. |
| vertices | Vertices | The two ends of the longer (major) axis. |
| c | Focal distance c | The distance from the center to each focus, √\|a² − b²\|. |
| e | Eccentricity | c divided by the longer semi-axis: 0 for a circle, near 1 for a long thin ellipse. |
| major | Major axis length | The longer axis, twice the longer semi-axis. |
| minor | Minor axis length | The shorter axis, twice the shorter semi-axis. |
| latus | Latus rectum | The chord through a focus at a right angle to the major axis: 2 × minor² ÷ major semi-axis. |

## Method

(x − h)²/a² + (y − k)²/b² = 1; c² = |a² − b²|; e = c ÷ max(a, b); area πab; perimeter 4·max(a, b)·E(e), computed by the arithmetic-geometric mean.

## Assumptions

- a is the semi-axis along x and b along y, so the longer one is the major axis.
- The ellipse’s axes are parallel to the x and y axes.
- The perimeter has no closed form; the arithmetic-geometric mean gives it to double precision.

## Worked examples

1. a = 3, b = 5, h = -2, k = -3 gives equation = (x + 2)²/9 + (y + 3)²/25 = 1, foci = (−2, −7), (−2, 1), vertices = (−2, −8), (−2, 2), c = 4, e = 0.8, area = 47.12389, perimeter = 25.526999. Source: OpenStax, Algebra and Trigonometry 2e, §12.1 The Ellipse (standard form, c² = a² − b², vertices and foci; vertices (−2, −8) and (−2, 2) with foci (−2, −7) and (−2, 1) give (x + 2)²/9 + (y + 3)²/25 = 1; area πab), https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-1-the-ellipse (retrieved 2026-10-05); perimeter: by the AGM, NIST Digital Library of Mathematical Functions, §19.8 (AGM: 19.8.1, 19.8.2, 19.8.6) and §19.9 (perimeter L(a, b) = 4aE(k), 19.9.9), https://dlmf.nist.gov/19.8 and https://dlmf.nist.gov/19.9 (retrieved 2026-10-05).
2. a = 2, b = 2 gives e = 0, c = 0, perimeter = 12.566371, area = 12.566371, foci = (0, 0), (0, 0). Source: OpenStax, Algebra and Trigonometry 2e, §12.1 The Ellipse (standard form, c² = a² − b², vertices and foci; vertices (−2, −8) and (−2, 2) with foci (−2, −7) and (−2, 1) give (x + 2)²/9 + (y + 3)²/25 = 1; area πab), https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-1-the-ellipse (retrieved 2026-10-05); NIST Digital Library of Mathematical Functions, §19.8 (AGM: 19.8.1, 19.8.2, 19.8.6) and §19.9 (perimeter L(a, b) = 4aE(k), 19.9.9), https://dlmf.nist.gov/19.8 and https://dlmf.nist.gov/19.9 (retrieved 2026-10-05).
3. a = 10, b = 6 gives c = 8, e = 0.8, latus = 7.2, area = 188.495559, perimeter = 51.053998, foci = (−8, 0), (8, 0). Source: OpenStax, Algebra and Trigonometry 2e, §12.1 The Ellipse (standard form, c² = a² − b², vertices and foci; vertices (−2, −8) and (−2, 2) with foci (−2, −7) and (−2, 1) give (x + 2)²/9 + (y + 3)²/25 = 1; area πab), https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-1-the-ellipse (retrieved 2026-10-05); NIST Digital Library of Mathematical Functions, §19.8 (AGM: 19.8.1, 19.8.2, 19.8.6) and §19.9 (perimeter L(a, b) = 4aE(k), 19.9.9), https://dlmf.nist.gov/19.8 and https://dlmf.nist.gov/19.9 (retrieved 2026-10-05).
4. a = 99, b = 1 gives perimeter = 396.110739. Source: NIST Digital Library of Mathematical Functions, §19.8 (AGM: 19.8.1, 19.8.2, 19.8.6) and §19.9 (perimeter L(a, b) = 4aE(k), 19.9.9), https://dlmf.nist.gov/19.8 and https://dlmf.nist.gov/19.9 (retrieved 2026-10-05).

## FAQ

### How do I find the area of an ellipse?

Multiply π by the two semi-axes: A = πab. An ellipse with semi-axes 3 and 5 has area 15π ≈ 47.12.

### How do I find the foci of an ellipse?

Find c from c² = a² − b², with a the longer semi-axis. The foci are c from the center along the longer axis. For semi-axes 3 (along x) and 5 (along y) centered at (−2, −3), c² = 25 − 9 = 16, so the foci are (−2, −7) and (−2, 1).

### Is there a formula for the perimeter of an ellipse?

Not in elementary functions. The perimeter is 4a times the complete elliptic integral E(e). The page computes it with Gauss’s arithmetic-geometric mean, which is exact to the last digit shown. Semi-axes 3 and 5 give about 25.527.

### What is the eccentricity of an ellipse?

e = c ÷ a, the focal distance over the longer semi-axis. It is 0 for a circle and close to 1 for a long, thin ellipse. Semi-axes 10 and 6 give c = 8 and e = 0.8.

### What is the standard form of an ellipse?

(x − h)²/a² + (y − k)²/b² = 1, with (h, k) the center, a the semi-axis along x and b along y. The longer of a and b is the semi-major axis.

### What is the latus rectum of an ellipse?

The chord through a focus at a right angle to the major axis. Its length is 2b² ÷ a, with a the longer and b the shorter semi-axis: 2 × 36 ÷ 10 = 7.2 for semi-axes 10 and 6.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §12.1 The Ellipse (standard form, c² = a² − b², vertices and foci, and area πab), CC BY 4.0. https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-1-the-ellipse (retrieved 2026-10-05)
- NIST Digital Library of Mathematical Functions, §19.8 Quadratic Transformations (the arithmetic-geometric mean, 19.8.1 and 19.8.2, and E(k) by it, 19.8.6). https://dlmf.nist.gov/19.8 (retrieved 2026-10-05)
- NIST Digital Library of Mathematical Functions, §19.9 Inequalities (the perimeter L(a, b) = 4aE(k), k² = 1 − b²/a², 19.9.9). https://dlmf.nist.gov/19.9 (retrieved 2026-10-05)