The slant height of the cone is the distance from the apex to a point in the perimeter of the base. If the cone is a right cone, the length of all the slant heights are the same.

$\,\left.\begin{array}{rcr} \mbox{slant height }\phantom{XXXX}\;\,\rightarrow\, & \;S_h\mbox{ = 5 cm }\; \\ \,\mbox{radius of the base}\phantom{XX} \rightarrow\, & R\,=\,\sqrt{3}\\ \mbox{Pythagorean theorem}\, \rightarrow\, & (S_h)^{\large 2}\,=\,H^{\large 2}\,+\,R^{\large 2}\; \\ \end{array} \right\}\;\Rightarrow\;$

$\;\Rightarrow\;5^{\large 2}\,=\,H^{\large 2}\,+\,(\sqrt{3})^{\large 2}\;\Leftrightarrow\;H\,=\,\sqrt{22} \mbox{ cm}$