Find the area $ A $ of a rhombus if side $a = 20$ and diagonal $d_2 = 36$.

$A = 72 \sqrt{ 19 }$

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 = 36 $ and $ a = 20 $ we have:

$$ d_1 ^ {\,2} + 36^2 = 4 \cdot 20^2 $$ $$ d_1 ^ {\,2} + 1296 = 1600 $$ $$ d_1 ^ {\,2} = 1600 - 1296 $$ $$ d_1 ^ {\,2} = 304 $$ $$ d_1 = \sqrt{ 304 } $$$$ d_1 = 4 \sqrt{ 19 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 4 \sqrt{ 19 } $ and $ d_2 = 36 $ we have:

$$ A = \dfrac{ 4 \sqrt{ 19 } \cdot 36 }{ 2 }$$$$ A = \dfrac{ 144 \sqrt{ 19 } }{ 2 } $$$$ A = 72 \sqrt{ 19 } $$

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