Find the area $ A $ of a rhombus if side $a = 5 \sqrt{ 5 }$ and diagonal $d_2 = 6 \sqrt{ 6 }$.

$A = 6 \sqrt{ 426 }$

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 = 6 \sqrt{ 6 } $ and $ a = 5 \sqrt{ 5 } $ we have:

$$ d_1 ^ {\,2} + \left(6 \sqrt{ 6 }\right)^2 = 4 \cdot \left(5 \sqrt{ 5 }\right)^2 $$ $$ d_1 ^ {\,2} + 216 = 500 $$ $$ d_1 ^ {\,2} = 500 - 216 $$ $$ d_1 ^ {\,2} = 284 $$ $$ d_1 = \sqrt{ 284 } $$$$ d_1 = 2 \sqrt{ 71 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 2 \sqrt{ 71 } $ and $ d_2 = 6 \sqrt{ 6 } $ we have:

$$ A = \dfrac{ 2 \sqrt{ 71 } \cdot 6 \sqrt{ 6 } }{ 2 }$$$$ A = \dfrac{ 12 \sqrt{ 426 } }{ 2 } $$$$ A = 6 \sqrt{ 426 } $$

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