Two rhombuses have sides with lengths of $2$ $2$ . If one rhombus has a corner with an angle of $\frac{π}{3}$ $pi/3$ and the other has a corner with an angle of $\frac{5 π}{8}$ $(5pi)/8$ , what is the difference between the areas of the rhombuses?

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Two rhombuses have sides with lengths of $2$ $2$ . If one rhombus has a corner with an angle of $\frac{π}{3}$ $pi/3$ and the other has a corner with an angle of $\frac{5 π}{8}$ $(5pi)/8$ , what is the difference between the areas of the rhombuses?

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Answer 1 · 2016-03-13T22:29:34

$D \Leftrightarrow e r e n c e = | A ⋄_{1} - A ⋄_{2} | = 2 \sqrt{3} - 2 \sqrt{2 + \sqrt{2}}$ $Difference=|Adiamond_1-Adiamond_2|= 2sqrt(3)-2sqrt(2+sqrt(2))$
$| A ⋄_{1} - A ⋄_{2} | = 2 (\sqrt{3} - \sqrt{2 + \sqrt{2}})$ $|Adiamond_1-Adiamond_2|= 2(sqrt(3)-sqrt(2+sqrt(2)))$
$| A ⋄_{1} - A ⋄_{2} | \approx 2 | 1.732 - 1.848 | = .231$ $|Adiamond_1-Adiamond_2| ~~ 2|1.732-1.848| = .231$

Explanation:

The easiest way to approach is using the following approach to finding the area of a parallelogram:
$A ⋄ = {\to s}_{1} \times {\to s}_{2} = | s_{1} \cdot s_{2} | sin θ$ $Adiamond= vecs_1 xx vecs_2 = |s_1*s_2| sin theta$ , where $θ$ $theta$ is the angle between the sides of the rhombuses. Since $s_{1} = s_{2}$ $s_1=s_2$ we write: $A ⋄ = ∣ ∣ s^{2} ∣ ∣ sin θ$ $Adiamond= |s^2| sin theta$ thus,
$A ⋄_{1} = 2^{2} sin (\frac{π}{3}) = 2 \sqrt{3}$ $Adiamond_1= 2^2 sin (pi/3) = 2sqrt(3)$
$A ⋄_{2} = 2^{2} sin (5 \frac{π}{8}) = 2 \sqrt{2 + \sqrt{2}}$ $Adiamond_2= 2^2 sin (5pi/8) = 2sqrt(2+sqrt(2))$
$D \Leftrightarrow e r e n c e = | A ⋄_{1} - A ⋄_{2} | = 2 \sqrt{3} - 2 \sqrt{2 + \sqrt{2}}$ $Difference=|Adiamond_1-Adiamond_2|= 2sqrt(3)-2sqrt(2+sqrt(2))$
$| A ⋄_{1} - A ⋄_{2} | = 2 (\sqrt{3} - \sqrt{2 + \sqrt{2}})$ $|Adiamond_1-Adiamond_2|= 2(sqrt(3)-sqrt(2+sqrt(2)))$
$| A ⋄_{1} - A ⋄_{2} | \approx 2 | 1.732 - 1.848 | = .231$ $|Adiamond_1-Adiamond_2| ~~ 2|1.732-1.848| = .231$

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Two rhombuses have sides with lengths of $2$ $2$ . If one rhombus has a corner with an angle of $\frac{π}{3}$ $pi/3$ and the other has a corner with an angle of $\frac{5 π}{8}$ $(5pi)/8$ , what is the difference between the areas of the rhombuses?

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Two rhombuses have sides with lengths of $2$ $2$ . If one rhombus has a corner with an angle of $\frac{π}{3}$ $pi/3$ and the other has a corner with an angle of $\frac{5 π}{8}$ $(5pi)/8$ , what is the difference between the areas of the rhombuses?