ConverterHub

Ellipse Area & Perimeter

How do I find the area of an ellipse?

Result

47.12
25.53

How it is calculated

The area is exact: A = π × a × b, where a and b are the two semi-axes, meaning half the full width and half the full height, in the same unit. An ellipse has no exact simple perimeter formula, so the tool uses Ramanujan's approximation: C ≈ π × (3(a + b) − √((3a + b)(a + 3b))). It is very accurate for most shapes. Both results use the entered unit, with area in square units. If a equals b the shape is a circle and the formulas reduce to πr² and 2πr. Both inputs must be above zero.

Worked example

An ellipse with semi-axes a = 5 and b = 3. Area: π × 5 × 3 = 47.12 square units. Circumference: 3 × (5 + 3) = 24; (3 × 5 + 3) × (5 + 3 × 3) = 18 × 14 = 252, whose root is about 15.875. So π × (24 − 15.875) ≈ 25.53 units.

Try this example →

Common mistakes

  • Entering the full width and height (diameters) instead of the semi-axes. Halve them first, otherwise area is 4 times too large.
  • Mixing units for a and b, such as centimetres for one and metres for the other.
  • Using π × (a + b) as the perimeter. That is a rough rule that is noticeably off for flat ellipses.
  • Forgetting that area is in square units while circumference is in linear units.

About this calculator

This calculator finds an ellipse's area, using the exact formula π × a × b where a and b are the two semi-axes, and its circumference, using an approximation (like Ramanujan's formula) since — unlike a circle — an ellipse has no simple closed-form formula for perimeter.

Ellipses come up in orbital mechanics (planets and satellites travel in elliptical orbits, not perfect circles), architecture and design (elliptical windows, tables, and stadium tracks), and engineering drawings of oval-shaped parts or ducts. Students and engineers plug in the two semi-axes to size a real elliptical shape — checking how much material or paint a table top needs, for instance, or working through a physics problem describing planetary orbits.

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