acalculator

What are the ellipse’s area and foci?

Type half the width and half the height of an ellipse, and its center if it is not at the origin. The ellipse calculator finds the area, the perimeter, the foci, the vertices, the eccentricity and the equation.

Your numbers

Area
47.1238898

An ellipse with semi-axes 3 and 5 has an area of 47.1238898 and a perimeter of 25.52699886.

Perimeter (circumference)
25.52699886
Equation
x²/9 + y²/25 = 1
Foci
(0, −4), (0, 4)
Vertices
(0, −5), (0, 5)
Focal distance c
4
Eccentricity
0.8
Major axis length
10
Minor axis length
6
Latus rectum
3.6

Area: 47.1238898. An ellipse with semi-axes 3 and 5 has an area of 47.1238898 and a perimeter of 25.52699886.

How to calculate

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

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.

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.

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Semi-axis along x (a) 3, Semi-axis along y (b) 5, Center x (h) -2, Center y (k) -3 gives Equation (x + 2)²/9 + (y + 3)²/25 = 1, Foci (−2, −7), (−2, 1), Vertices (−2, −8), (−2, 2), Focal distance c 4, Eccentricity 0.8, Area 47.12389, Perimeter (circumference) 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. Semi-axis along x (a) 2, Semi-axis along y (b) 2 gives Eccentricity 0, Focal distance c 0, Perimeter (circumference) 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. Semi-axis along x (a) 10, Semi-axis along y (b) 6 gives Focal distance c 8, Eccentricity 0.8, Latus rectum 7.2, Area 188.495559, Perimeter (circumference) 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. Semi-axis along x (a) 99, Semi-axis along y (b) 1 gives Perimeter (circumference) 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)

How it works

Type a, the semi-axis along x, and b, the semi-axis along y, and the center (h, k). Let A = max(a, b) be the longer semi-axis and B = min(a, b) the shorter one.

  • Equation: (x − h)²/a² + (y − k)²/b² = 1.
  • Focal distance: c² = A² − B² = |a² − b²|, c = √(c²).
  • Foci and vertices: on the longer axis. When a ≥ b they are (h ± c, k) and (h ± a, k); when b > a they are (h, k ± c) and (h, k ± b). A circle (a = b) has both foci at the center.
  • Eccentricity: e = c ÷ A.
  • Area: πab.
  • Axes: major 2A, minor 2B; latus rectum 2B² ÷ A.
  • Perimeter: L = 4A·E(e), where E is the complete elliptic integral of the second kind. With the arithmetic-geometric mean a₀ = A, g₀ = B, aₙ₊₁ = (aₙ + gₙ) ÷ 2, gₙ₊₁ = √(aₙgₙ), cₙ = (aₙ₋₁ − gₙ₋₁) ÷ 2 for n ≥ 1, and c₀² = A² − B², this is L = 2π(A² − Σₙ₌₀ 2ⁿ⁻¹cₙ²) ÷ M, where M is the common limit of aₙ and gₙ. The page scales A to 1 and stops when a term 2ⁿ⁻¹cₙ² falls to 10⁻¹⁷ or below (at most 40 steps), then multiplies by A.

Rules

  • a and b are from 10⁻¹² to 10¹²; h and k are from −10¹² to 10¹² and are 0 when left empty.
  • The axes of the ellipse are parallel to the x and y axes.

Output format. a², b² and c² are exact from the typed decimals; the equation, the vertices and (when c² is the square of a fraction) the foci are exact, rounded once to 10 significant figures. Otherwise the foci, c, e, the area, the latus rectum and the perimeter are double-precision numbers shown to 10 significant figures, with the true minus sign (−).

Worked examples by hand

Semi-axes 3 (x) and 5 (y), center (−2, −3). The longer axis is vertical. Equation (x + 2)²/9 + (y + 3)²/25 = 1. c² = 25 − 9 = 16, c = 4, foci (−2, −7), (−2, 1), vertices (−2, −8), (−2, 2), e = 4 ÷ 5 = 0.8, area 15π = 47.1239. Perimeter by the AGM from (5, 3): 25.5270.

A circle, a = b = 2. c = 0, e = 0, both foci at (0, 0). Area 4π = 12.5664; the AGM stops at once (c₀ = 0), so L = 2π × 4 ÷ 2 = 4π = 12.5664.

Semi-axes 10 and 6. c = √(100 − 36) = 8, e = 0.8, foci (−8, 0) and (8, 0), latus rectum 2 × 36 ÷ 10 = 7.2, area 60π = 188.4956, perimeter 51.0540.

Semi-axes 99 and 1. A long, thin ellipse: perimeter 396.1107, close to but above 4 × 99 = 396. A numerical integral of 4 × 99 × ∫₀^(π/2) √(1 − e² sin²θ) dθ gives the same value.

Other questions people ask

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.