The parametric equations that represent the ellipse are and , where .

Consider an ellipse and make it into equal parts.

Let us consider the first quadrant, which is from to ,the curve starts at goes clockwise to trace out ellipse.

Since it has the horizantal major axis.

Consider .

Diffrentiate with respective to .

Consider .

Diffrentiate with respective to .

Find the length of the curve.

Theorem :

If a curve is described by the parametric equations  and , .

Then the length of the curve is

Where and .

As we considered the first quadrant, which is from to

Therefore and .

Use symmetry change the limits to first quadrant only, multiplying by .

From the trignometric identity :

.

For an ellicpse the distance from the center to the focus is , which is given by 

By squaring on both sides :

The eccentricity of the ellicpse is

Substiute the value .

Since ,  .

The total length of the ellicpse is .