STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $P = 64\, \text{cm}$ we have:

$$ 64\, \text{cm} = 4 \cdot a $$ $$ a = \dfrac{ 64\, \text{cm} }{ 4 } $$ $$ a = 16\, \text{cm} $$

STEP 2: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

$$ d_1^2 + d_2^2 = 4 \cdot a^2 $$

After substituting $d_2 = 8 \sqrt{ 5 }\, \text{cm}$ and $a = 16\, \text{cm}$ we have:

$$ d_1 ^ {\,2} + \left( 8 \sqrt{ 5 }\, \text{cm} \right)^{2} = 4 \cdot \left( 16\, \text{cm} \right)^{2} $$ $$ d_1 ^ {\,2} + 320\, \text{cm}^2 = = 1024\, \text{cm}^2 $$ $$ d_1 ^ {\,2} = = 1024\, \text{cm}^2 - 320\, \text{cm}^2 $$ $$ d_1 ^ {\,2} = 704\, \text{cm}^2 $$ $$ d_1 = \sqrt{ 704\, \text{cm}^2 } $$$$ d_1 = 8 \sqrt{ 11 }\, \text{cm} $$

STEP 3: find area $ A $

To find area $ A $ use formula:

$$ A = \dfrac{ d_1 \cdot d_2 }{ 2 } $$

After substituting $d_1 = 8 \sqrt{ 11 }\, \text{cm}$ and $d_2 = 8 \sqrt{ 5 }\, \text{cm}$ we have:

$$ A = \dfrac{ 8 \sqrt{ 11 }\, \text{cm} \cdot 8 \sqrt{ 5 }\, \text{cm} }{ 2 }$$$$ A = \dfrac{ 64 \sqrt{ 55 }\, \text{cm}^2 }{ 2 } $$$$ A = 32 \sqrt{ 55 }\, \text{cm}^2 $$

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