Find the area $ A $ of a rhombus if perimeter $P = 80$ and diagonal $d_1 = 36$.

$A = 72 \sqrt{ 19 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 80 $ we have:

$$ 80 = 4 \cdot a $$ $$ a = \dfrac{ 80 }{ 4 } $$ $$ a = 20 $$

STEP 2: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ d_1 = 36 $ and $ a = 20 $ we have:

$$ 36^2 + d_2^2 = 4 \cdot 20^2 $$ $$ 1296 + d_2^2 = 1600 $$ $$ d_2^2 = 1600 - 1296 $$ $$ d_2^2 = 304 $$ $$ d_2 = \sqrt{ 304 } $$$$ d_2 = 4 \sqrt{ 19 } $$

STEP 3: find area $ A $

To find area $ A $ use formula:

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

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

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

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