The platonic solid built with 5 equilateral triangles to each vertex is an icosahedron (20 faces). To make its dual with the Platonic Solids kit, follow the instructions in this series linked here. 

If you imagine the structure as of its faces and unfold it, you get its net. 

As part of the Platonic Solids series, this model has clearly defined colors outlining the solid itself, scaffolding in black, and its dual in white.