Cross-sectional area of a helix

I'm trying to find a general formula for the cross-sectional area of a 3D circular constant helix if it is sliced parallel to its axis.

The shape formed should be an ellipse, but I need the input parameters to be in terms of typical helical parameters (axial pitch, helix diameter, etc.).

Picture a spring sitting flat on a table. The end of it was sliced so it can sit flat, but what is the area of the flat?

My colleague found the formula for how much longer the helix will be than it is tall, and then used this ratio to scale the cross section of the circle. Is this a valid approach?

I appreciate any ideas that you have. I have a minor in mathematics and feel super dumb for not being able to figure this out. (Worried)

Thank you very much for your time! Have a great day.