Find the area $ A $ of a rhombus if perimeter $P = 32$ and angle $\alpha = 45^o$.

$A = 32 \sqrt{ 2 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 32 $ we have:

$$ 32 = 4 \cdot a $$ $$ a = \dfrac{ 32 }{ 4 } $$ $$ a = 8 $$

STEP 2: find area $ A $

To find area $ A $ use formula:

$$ A = \dfrac{ a \cdot a \cdot \sin( \alpha ) }{ 1 } $$

After substituting $ a = 8 $ , $ a = 8 $ and $ \alpha = 45^o $ we have:

$$ A = \dfrac{ 8 \cdot 8 \cdot \sin( 45^o ) }{ 1 } $$ $$ A = \dfrac{ 8 \cdot 8 \cdot \frac{\sqrt{ 2 }}{ 2 } }{ 1 } $$ $$ A = \dfrac{ 64 \cdot \frac{\sqrt{ 2 }}{ 2 } }{ 1 } $$ $$ A = \dfrac{ 32 \sqrt{ 2 } }{ 1 } $$ $$ A = 32 \sqrt{ 2 } $$

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