Diagonals of the rhombus

How long are the diagonals e, f in the diamond, if its side is 5 cm long and its area is 20 cm2?

Result

e =  4.472 cm
f =  8.944 cm

Solution:

a=5 cm S=20 cm2  S=ah h=S/a=20/5=4 cm  h=a⋅ sin⁡(A) A=180∘π⋅arcsin⁡(h/a)=180∘π⋅arcsin⁡(4/5)≐53.1301 ∘  e=a⋅ 2−2⋅ cos⁡(A∘→ rad)=a⋅ 2−2⋅ cos⁡(A∘⋅ π180 )=5⋅ 2−2⋅ cos⁡(53.130102354156∘⋅ 3.1415926180 )=4.47214=4.472 cma=5 \ \text{cm} \ \\ S=20 \ \text{cm}^2 \ \\ \ \\ S=a h \ \\ h=S / a=20 / 5=4 \ \text{cm} \ \\ \ \\ h=a \cdot \ \sin(A) \ \\ A=\dfrac{ 180^\circ }{ \pi } \cdot \arcsin(h/a)=\dfrac{ 180^\circ }{ \pi } \cdot \arcsin(4/5) \doteq 53.1301 \ ^\circ \ \\ \ \\ e=a \cdot \ \sqrt{ 2 - 2 \cdot \ \cos( A ^\circ \rightarrow\ \text{rad}) }=a \cdot \ \sqrt{ 2 - 2 \cdot \ \cos( A ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ ) }=5 \cdot \ \sqrt{ 2 - 2 \cdot \ \cos( 53.130102354156 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ ) }=4.47214=4.472 \ \text{cm}
f=a⋅ 2+2⋅ cos⁡(A∘→ rad)=a⋅ 2+2⋅ cos⁡(A∘⋅ π180 )=5⋅ 2+2⋅ cos⁡(53.130102354156∘⋅ 3.1415926180 )=8.94427=8.944 cmf=a \cdot \ \sqrt{ 2 + 2 \cdot \ \cos( A ^\circ \rightarrow\ \text{rad}) }=a \cdot \ \sqrt{ 2 + 2 \cdot \ \cos( A ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ ) }=5 \cdot \ \sqrt{ 2 + 2 \cdot \ \cos( 53.130102354156 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ ) }=8.94427=8.944 \ \text{cm}

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