Is there a way to cut up an equilateral triangle and reorganize the parts to get a square of the same surface area?

This is known as Haberdasher's Problem.  A web search will get you the solution.

A search on Mathematical Disections gets further more general info.

Well there is of course but working out the best cuts would take more time than i have right now.

If you let the side lengths of the equilateral triangle be 2 units then the height will be $$\sqrt3$$  units.

You can multiply these by any constant if you want different side lengths.

The area of the triangle will be      $$A=(1/2)*b*h = 0.5*2*\sqrt3 = \sqrt3$$

The are of a square is    $$l*l$$  so

$$\\l*l=\sqrt3\\
l^2=3^{1/2}\\
l=3^{1/4}\qquad$or if you prefer $ l=\sqrt[4]{3}\;\; units$$

so if the side of the triangle is 2k units then the side of the square will be   $$\sqrt[4]{3}\times k\;\; units$$

Thanks Bertie :)