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

(1) P = \pi (a + b)
Where P = is the perimeter or circumference
a = is the semi-major axis
b = is the semi-minor axis

 

(b) When the ellipse differs from a circle (i.e., when there is a 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 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