Kepler's second law of motion Planets sweeps out equal areas in equal times. 

Is the area being calculated here the area of a triangle? You draw a straight line from the center of the sun to the earth at point A. The Earth moves to Point B and you have another straight line from point A to point B and the finally another straight line from point B to the Sun? Even though the Earth makes an arc, I guess we make it a straight line.

I guess my question is whether I have understood the second law correctly? That the area of a triangle was being calculated in an elliptical orbit, no matter how elliptical the orbit is.
That is not correct.

The area is the total area between the two radius lines, so there is a curved side.

Imagine you have two points almost 180 degrees from each other. Using just a triangle, the area is close to zero. Now, two points placed closely together can have the same area in between them. Then you have an equal are, but not equal time, so your hypothesis can not be correct.