Kepler’s ellipse perimeter approximations

In 1609, Kepler remarked that the perimeter of an ellipse with semiaxes a and b could be approximated either as

P ≈ 2π(ab)½


P ≈ π(a + b).

In other words, you can approximate the perimeter of an ellipse by the circumference of a circle of radius r where r is either the geometric mean or arithmetic mean of the semi-major and semi-minor axes.

Comparing ellipse with circles of approximately the same perimeter

How good are these approximations, particularly when a and b are roughly equal? Which one is better?

When can choose our unit of measurement so that the semi-minor axis b equals 1, then plot the error in the two approximations as a increases.

Exact and approximation ellipse perimeter

We see from this plot that both approximations give lower bounds, and that arithmetic mean is more accurate than geometric mean.

Incidentally, if we used the geometric mean of the semi-axes as the radius of a circle when approximating the area then the results would be exactly correct. But for perimeter, the arithmetic mean is better.

Ellipse approximation relative error

Next, if we just consider ellipses in which the semi-major axis is no more than twice as long as the semi-minor axis, the arithmetic approximation is within 2% of the exact value and the geometric approximation is within 8%. Both approximations are good when ab.

The next post goes into more mathematical detail, explaining why Kepler’s approximation behaves as it does and giving ways to improve on it.

More ellipse posts

2 thoughts on “Kepler’s ellipse perimeter approximations

  1. Marnix Klooster


    This seems a case for : P ≈ π(a + b) is more clearly about the arithmetic mean if written as P ≈ 2π(a + b)/2. And that also better brings out the symmetry between these two approximations.

    Note that several time (mostly in images), the _harmonic_ mean is mentioned?

  2. back in trhe early 60’s in engineering drawing, we learned the four center ellipse (drawing) approximation I wonder how the lentgh of this approximation compares to the correct value Too lazy to figure it out right now maybe later

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