Find the side $ a $ of a rhombus if area $A = 30$ and diagonal $d_1 = \frac{ 693 }{ 100 }$.

$a = \frac{ 13 \sqrt{ 388322881}}{ 46200 }$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ A = 30 $ and $ d_1 = \frac{ 693 }{ 100 } $ we have:

$$ 30 = \dfrac{ \frac{ 693 }{ 100 } \cdot d_2 }{ 2 } $$$$ 30 \cdot 2 = \frac{ 693 }{ 100 } \cdot d_2 $$$$ 60 = \frac{ 693 }{ 100 } \cdot d_2 $$$$ d_2 = \dfrac{ 60 }{ \frac{ 693 }{ 100 } } $$$$ d_2 = \frac{ 2000 }{ 231 } $$

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d_1 = \frac{ 693 }{ 100 } $ and $ d_2 = \frac{ 2000 }{ 231 } $ we have:

$$ \left(\frac{ 693 }{ 100 }\right)^2 + \left(\frac{ 2000 }{ 231 }\right)^2 = 4 \cdot a^2 $$ $$ \frac{ 480249 }{ 10000 } + \frac{ 4000000 }{ 53361 } = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = \frac{ 65626566889 }{ 533610000 } $$ $$ a^2 = \frac{ \frac{ 65626566889 }{ 533610000 } }{ 4 } $$ $$ a^2 = \frac{ 65626566889 }{ 2134440000 } $$ $$ a = \sqrt{ \frac{ 65626566889 }{ 2134440000 } } $$$$ a = \frac{ 13 \sqrt{ 388322881}}{ 46200 } $$

Rhombus Calculator