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Ellipse Calculator

Enter an oval's half-width and half-height to get its area, exact perimeter, how stretched it is, its key points and its equation.

Level: Grades 11–12

Printable worksheet

Ellipse Calculator

Know the answer you need? Pick an input to work backward.

Inputs

Half the ellipse's full width, measured side to side.

Half the full height, measured up and down.

Result

Area

47.1239 square units

An ellipse with semi-axes 5 and 3 has area 47.1239, perimeter 25.527 and eccentricity 0.8.

Perimeter (circumference)
25.527
Perimeter, Ramanujan's approximation
25.527
Eccentricity (e)
0.8
Focal distance (c)
4
Foci
(−4, 0) and (4, 0)
Vertices (ends of the major axis)
(−5, 0) and (5, 0)
Co-vertices (ends of the minor axis)
(0, −3) and (0, 3)
Semi-latus rectum
1.8
Standard-form equation
x²/25 + y²/9 = 1
Orientation
Major axis horizontal

See the step-by-step solution ↓

Step-by-step solution

  1. Find the area: π × a × b. It is the circle formula, with the radius squared swapped for the two half-widths multiplied

    Area=π×5×3=47.12389

  2. The horizontal axis is the longer (major) one. Find c, how far each focus sits from the center, with the right-triangle rule

    c=52−32=4

  3. Divide c by the longer half-axis to get e, the eccentricity. It shows how stretched the shape is, where 0 is a circle

    e=45=0.8

  4. The perimeter has no everyday formula. So keep taking the usual average and the geometric average of the two half-axes until they agree. That shared value is M

    AGM(5, 3)=3.936236

  5. Add up the gaps left over from each round. This sum gives the exact perimeter P

    P=25.526999

  6. As a check, Ramanujan's shortcut formula, π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))) with h = (a − b)² ÷ (a + b)², gives almost the same answer

    P=≈ 25.526999

  7. Write the ellipse's equation in standard form, with its center at (h, k)

    x²/25 + y²/9 = 1

  8. The two foci sit on the longer axis, a distance c on each side of the center

    (−4, 0) and (4, 0)

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find the area, the distance around and the key points of any oval shape (an ellipse). An ellipse is a stretched circle. Pick two fixed points inside it, called the foci. For every point on the curve, the distances to those two points add up to the same total. Two measurements fix its size and shape. The semi-major axis is half the longest width, and the semi-minor axis is half the shortest. Orbits, oval tables and circles seen at an angle are all ellipses. So is a running track seen from above.

Enter the horizontal and vertical semi-axes (half the width and half the height). If the center isn't at the origin, the point (0, 0), enter the center too. You get the area and the perimeter, worked out to full precision. You also get the eccentricity, a number for how stretched the shape is. Finally, you see the points that mark the foci and the ends of both axes (the vertices and co-vertices). The ellipse's equation in standard form comes last.

Where it's used

Why this math matters outside the classroom:

  • Planet orbits. Astronomers describe Earth's orbit as an ellipse with an eccentricity of about 0.017, so close to a circle it looks round on paper.
  • Whispering galleries. In an elliptical room a whisper at one focus is heard at the other, so designers locate the foci with c² = a² − b².
  • Oval tables. An oval table 6 ft long and 4 ft wide has a top area of π × 3 × 2, about 18.8 square feet.
  • Garden ovals. For an oval bed 10 ft long and 6 ft wide, gardeners drive stakes at the foci, 4 ft either side of center, and trace around them with a string loop.
  • Lamp reflectors. Lamp makers shape elliptical reflectors so light from a bulb at one focus gathers at a point on the other focus.

How to use it

  1. Measure the full width and full height of the ellipse, and halve each one. Enter half the width as a and half the height as b.
  2. If the ellipse is centered somewhere other than (0, 0), enter the center's position: h across and k up. Otherwise, leave them at 0.
  3. Read the area at the top, followed by the perimeter, eccentricity and the key points.
  4. Open the working to see the formulas and how the perimeter is found.

What the inputs mean

Horizontal semi-axis (a)
Half the width, measured side to side (the x-direction). It can be larger or smaller than b. Whichever is larger is the semi-major axis.
Vertical semi-axis (b)
Half the height, measured up and down (the y-direction). If a and b are equal, the ellipse is a circle.
Center x-coordinate (h)
How far across the center sits (its x-coordinate). It moves the foci, the vertices and the equation, but it doesn't change the size or shape.

Formula and method

Area = π × a × b
c = √(A² − B²),   e = c ÷ A   (A = larger semi-axis, B = smaller)
P = (2π ÷ M) × (A² − Σ 2ⁿ⁻¹ cₙ²),   M = AGM(A, B)
P ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))),   h = (a − b)² ÷ (a + b)²
(x − h)² ÷ a² + (y − k)² ÷ b² = 1

Here a is the horizontal semi-axis and b is the vertical one. A is whichever is larger and B is the smaller. π is pi, about 3.14159. The area formula is the circle's πr², with r² swapped for a × b. The focal distance c is how far each focus sits from the center. It comes from the Pythagorean theorem, the right-triangle rule. The eccentricity e = c ÷ A runs from 0 for a circle up toward 1 for a very flat ellipse. In the last line, (h, k) is the center, and x and y are any point on the curve.

Unlike the area, the perimeter (P) has no formula made of everyday math steps. Mathematicians call it an elliptic integral. So the calculator uses the Gauss–Kummer method, which works in rounds. Each round takes two kinds of average of the semi-axes: the usual average, and the geometric average (the square root of their product). After a few rounds the two averages agree. That shared value is M, the arithmetic–geometric mean, or AGM. Along the way, the gaps between the averages (the cₙ terms, one per round n) are added up into a correction sum, Σ. A few rounds land on the exact value. Ramanujan's well-known shortcut formula is shown next to it. In that formula, h is a measure of how different a and b are, not the center. For ordinary shapes it agrees to many decimal places.

Worked example

An oval table top is 1.8 m long and 1.2 m wide. How much surface does it have, and how much edge banding is needed to go around it?

  1. Semi-axes: a = 1.8 ÷ 2 = 0.9 m and b = 1.2 ÷ 2 = 0.6 m.
  2. Area = π × 0.9 × 0.6 ≈ 1.6965 m².
  3. Focal distance c = √(0.9² − 0.6²) = √0.45 ≈ 0.6708 m, so the eccentricity is 0.6708 ÷ 0.9 ≈ 0.7454.
  4. The perimeter from the arithmetic–geometric mean method is about 4.7596 m. Ramanujan's shortcut gives the same value to nine decimal places.

Result: The table top covers about 1.6965 m², and about 4.7596 m of edge banding goes around it.

Understanding the result

Eccentricity sums up the shape. Near 0, the ellipse looks almost like a circle. Around 0.8, it's clearly oval, and near 1 it becomes long and thin. It doesn't depend on size. So two ellipses with the same eccentricity are scaled copies of each other.

The foci sit inside the ellipse on its longer axis. For any point on the curve, the distances to the two foci add up to twice the semi-major axis. That's why you can draw an ellipse with a loop of string around two pins.

Assumptions and limitations

  • The ellipse lines up with the x- and y-axes (side to side and up and down). Tilted, rotated ellipses aren't handled.
  • All lengths use one unit, and the area is in that unit squared.
  • It doesn't work backward to find the semi-axes from a perimeter, or from a general second-degree equation (one with x², y² and xy terms).
  • Rotated ellipses and three-dimensional ellipsoids (egg or football shapes) aren't covered.

Frequently asked questions

Why is there no simple formula for the perimeter of an ellipse?

Measuring the curve of an ellipse leads to an elliptic integral. It can't be written with everyday math functions like roots, powers or sines. That's why shortcuts like Ramanujan's exist. It's also why this calculator uses a fast method that repeats a step until it reaches the exact value.

Which is the major axis?

Whichever semi-axis is longer. If a > b, the major axis runs side to side, and the foci sit left and right of the center. If b > a, it runs up and down, and the foci sit above and below.

How accurate is Ramanujan's approximation?

For fairly oval shapes, it matches the true perimeter to around nine or ten significant figures (digits that count). The error only grows for extremely flat ellipses. Even then, it stays well under one part in ten thousand.

What is the semi-latus rectum?

It's half the width of the ellipse measured straight through a focus, at a right angle to the major axis. It equals B² ÷ A. It shows up in the math of orbits and in the polar equation of an ellipse (the version that uses angle and distance).

How this calculator is checked

Method: Area πab; c² = A² − B²; perimeter by the Gauss–Kummer AGM series (checked against numeric integration); Ramanujan's second approximation. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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