Find the area $ A $ of a rhombus if side $a = \sqrt{ 5 }$ and diagonal $d_1 = \sqrt{ 6 }$.

$A = \sqrt{ 21 }$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ d_1 = \sqrt{ 6 } $ and $ a = \sqrt{ 5 } $ we have:

$$ \left( \sqrt{ 6 } \right)^{ 2 } + d_2^2 = 4 \cdot \left( \sqrt{ 5 } \right)^{ 2 } $$ $$ 6 + d_2^2 = 20 $$ $$ d_2^2 = 20 - 6 $$ $$ d_2^2 = 14 $$ $$ d_2 = \sqrt{ 14 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = \sqrt{ 6 } $ and $ d_2 = \sqrt{ 14 } $ we have:

$$ A = \dfrac{ \sqrt{ 6 } \cdot \sqrt{ 14 } }{ 2 }$$$$ A = \dfrac{ 2 \sqrt{ 21 } }{ 2 } $$$$ A = \sqrt{ 21 } $$

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