# 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:

$a=5 \ \text{cm} \ \\ S=20 \ \text{cm}^2 \ \\ \ \\ S=a h \ \\ h=S / a=20 / 5=4 \ \text{cm} \ \\ \ \\ h=a \cdot \ \sin(A) \ \\ A=\dfrac{ 180^\circ }{ \pi } \cdot \arcsin(h/a)=\dfrac{ 180^\circ }{ \pi } \cdot \arcsin(4/5) \doteq 53.1301 \ ^\circ \ \\ \ \\ e=a \cdot \ \sqrt{ 2 - 2 \cdot \ \cos A ^\circ }=a \cdot \ \sqrt{ 2 - 2 \cdot \ \cos 53.1301023542^\circ \ }=5 \cdot \ \sqrt{ 2 - 2 \cdot \ \cos 53.1301023542^\circ \ }=a \cdot \ \sqrt{ 2 - 2 \cdot \ 0.6 }=4.47214=4.472 \ \text{cm}$
$f=a \cdot \ \sqrt{ 2 + 2 \cdot \ \cos A ^\circ }=a \cdot \ \sqrt{ 2 + 2 \cdot \ \cos 53.1301023542^\circ \ }=5 \cdot \ \sqrt{ 2 + 2 \cdot \ \cos 53.1301023542^\circ \ }=a \cdot \ \sqrt{ 2 + 2 \cdot \ 0.6 }=8.94427=8.944 \ \text{cm}$ Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

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