The 'formula', surely, is simply pythagoras for right-angled triangles?

http://www.cleavebooks.co.uk/scol/calrtri.htm

The axis of the Earth is a vertical from point A. The radius at any point of latitude will equal side b, as it will be a line parallel to side b, connecting C with a point on the vertical above A. 

For 10 degrees N or S, imagine the Earth viewed from the side. Hypotenuse is Earth radius of 6,371km (side c), angle A = 10 degrees, put these into the calculator, and side b results at 6,270km

For 20 degrees, b = 5,990 km

Repeat, repeat for 30 to 80 degrees

Also check this site https://www.easyunitconverter.com/right-triangle-calculator