Find the area $ A $ of a rhombus if perimeter $P = 64$ and diagonal $d_2 = 8 \sqrt{ 5 }$.

$A = 32 \sqrt{ 55 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 64 $ we have:

$$ 64 = 4 \cdot a $$ $$ a = \dfrac{ 64 }{ 4 } $$ $$ a = 16 $$

STEP 2: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ d_2 = 8 \sqrt{ 5 } $ and $ a = 16 $ we have:

$$ d_1 ^ {\,2} + \left(8 \sqrt{ 5 }\right)^2 = 4 \cdot 16^2 $$ $$ d_1 ^ {\,2} + 320 = 1024 $$ $$ d_1 ^ {\,2} = 1024 - 320 $$ $$ d_1 ^ {\,2} = 704 $$ $$ d_1 = \sqrt{ 704 } $$$$ d_1 = 8 \sqrt{ 11 } $$

STEP 3: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 8 \sqrt{ 11 } $ and $ d_2 = 8 \sqrt{ 5 } $ we have:

$$ A = \dfrac{ 8 \sqrt{ 11 } \cdot 8 \sqrt{ 5 } }{ 2 }$$$$ A = \dfrac{ 64 \sqrt{ 55 } }{ 2 } $$$$ A = 32 \sqrt{ 55 } $$

Rhombus Calculator