# 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