Perimeter of an Ellipse

The perimeter or circumference of an ellipse is difficult to determine accurately. Various approximation formulas are given for finding the perimeter of an ellipse.

(a) If the ellipse is very nearly of the shape of a circle (i.e., if the major and minor axes are nearly equal), then perimeter is given by

(1) $P = \pi (a + b)$
Where $P =$perimeter or circumference
$a =$ Semi-major axis
$b =$ Semi-minor axis

(b) When the ellipse differs from a circle (i.e., when there is considerable difference between the major and minor axes), either of the followings may be used:

(2) $P = \pi \left[ {\frac{3}{2}(a + b) - \sqrt {ab} } \right]$

(3) $P = \pi \sqrt {2({a^2} + {b^2})}$

The result obtained by using formula (2) is too small and the result from formula (3) is too large. However, the mean or average of these results is very approximately correct.

Example:

Find the perimeter of the ellipse whose major axis is $18$cm, and whose minor axis is $6$cm.

Solution:
Here $a = 9$,   $b = 3$

From (1):   $P = \pi (a + b) = 3.1516(9 + 3) = 37.699$cm

From (2):   $P = \pi \left[ {\frac{3}{2}(a + b) - \sqrt {ab} } \right] = 3.1416\left[ {\frac{3}{2}(9 + 3) - \sqrt {9 \times 3} } \right] = 40.225$cm

From (3):   $P = \pi \sqrt {2({a^2} + {b^2})} = 3.1416\sqrt {2({9^2} + {3^2})} = 41.48$cm