To find incircle radius $ r $ use formula:
$$ r = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$After substituting $a = 1\, \text{cm}$ we have:
$$ r = \dfrac{ \sqrt{ 3 } \cdot 1\, \text{cm} }{ 2 } $$$$ r = \dfrac{ \sqrt{ 3 }\, \text{cm} }{ 2 } $$ $$ r = \frac{\sqrt{ 3 }}{ 2 }\, \text{cm} $$