Here are collected properties of regular icosahedron which are useful for students of mathematics or mathematics teachers who can prepare exercises for talented students in elementary or middle school. The initial section describes the basic properties of regular polyhedra: tetrahedron, cube, dodecahedron, octahedron and of course icosahedron. We have proven that there are only five regular or platonic solids and have verified Euler's polyhedron formula for them. Then we focused on selected polyhedron. To calculate the volume we need the golden number which is a positive solution of the Fibonacci equation. In the last part we have made planar graph for icosahedron and have told something about Hamilton's trip around the world (Icosian game) and explore symmetry of icosahedron.
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