30-60-90 Polypuzzle

Can you re-arrange the pieces of the puzzle to form a rectangle by sliding the pieces without rotating them? Now can you re-arrange the pieces to form an equilateral triangle by flipping the pieces numbered $2$ and $5$ and moving them into new positions?

You can assume that pieces $1$ and $5$ each have a side of length one unit, that the pieces as shown form a perfect square of area one square unit and that they do fit together to form a perfect equilateral triangle of the same area. 

Calculate the length $t$ of the edge of piece $3$ and then calculate the lengths of all the other edges giving answers correct to $3$ significant figures. You can use the interactivity below to explore how the pieces fit together.