Area enclosed by an ellipse

The number of square units it takes to exactly fill the interior of an ellipse.
Try this Drag the orange dots to move and resize the ellipse. As the shape and size of the ellipse changes, the area is recalculated.

An ellipse is actually a line, one that connects back to itself making a loop. Imagine the ellipse to be a loop of string. The string itself has no area, but the space inside the loop does. So strictly speaking an ellipse has no area.

However, when we say "the area of an ellipse" we really mean the area of the space inside the ellipse. If you were to cut a elliptical disk from a sheet of paper, the disk would have an area, and that it what we mean here.

Area Formula

Given the major radius and minor radius* of an ellipse, the area inside it can be calculated using the formula where:
j  is the major radius of the ellipse
n  is the minor radius of the ellipse
*Also known as the semi-major and semi-minor axis of the ellipse

Relation to a Circle

A circle is just a special case of an ellipse where both axes are the same length. In the figure above, carefully adjust the ellipse by dragging any orange dot until the ellipse becomes a circle. You will see that because the major and minor radii are the same, the area is the familiar 'Pi times radius squared'.

Things to try

  1. In the figure above, click on "hide details"
  2. Drag one of the four orange dots on the edge of the ellipse to make a random-size ellipse.
  3. Estimate the area enclosed by the ellipse just by looking at the squares inside it
When you done click "show details" to see how close you got.

Other ellipse topics