Find the area $ A $ of a rhombus if side $a = 16$ and diagonal $d_2 = 8$.

$A = 32 \sqrt{ 15 }$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ d_2 = 8 $ and $ a = 16 $ we have:

$$ d_1 ^ {\,2} + 8^2 = 4 \cdot 16^2 $$ $$ d_1 ^ {\,2} + 64 = 1024 $$ $$ d_1 ^ {\,2} = 1024 - 64 $$ $$ d_1 ^ {\,2} = 960 $$ $$ d_1 = \sqrt{ 960 } $$$$ d_1 = 8 \sqrt{ 15 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 8 \sqrt{ 15 } $ and $ d_2 = 8 $ we have:

$$ A = \dfrac{ 8 \sqrt{ 15 } \cdot 8 }{ 2 }$$$$ A = \dfrac{ 64 \sqrt{ 15 } }{ 2 } $$$$ A = 32 \sqrt{ 15 } $$

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