Find the area $ A $ of a rhombus if side $a = 16$ and diagonal $d_2 = 4$.

$A = 24 \sqrt{ 7 }$

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 = 4 $ and $ a = 16 $ we have:

$$ d_1 ^ {\,2} + 4^2 = 4 \cdot 16^2 $$ $$ d_1 ^ {\,2} + 16 = 1024 $$ $$ d_1 ^ {\,2} = 1024 - 16 $$ $$ d_1 ^ {\,2} = 1008 $$ $$ d_1 = \sqrt{ 1008 } $$$$ d_1 = 12 \sqrt{ 7 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 12 \sqrt{ 7 } $ and $ d_2 = 4 $ we have:

$$ A = \dfrac{ 12 \sqrt{ 7 } \cdot 4 }{ 2 }$$$$ A = \dfrac{ 48 \sqrt{ 7 } }{ 2 } $$$$ A = 24 \sqrt{ 7 } $$

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