# Two rhombuses have sides with lengths of #15 #. If one rhombus has a corner with an angle of #(3pi)/8 # and the other has a corner with an angle of #(11pi)/12 #, what is the difference between the areas of the rhombuses?

Difference between areas of the two rhombuses is 44.3763

Area of rhombus

Where

In this case we will use the formula Area = a * h.

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The area of a rhombus can be calculated using the formula: ( \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 ).

Given that both rhombuses have sides with lengths of 15, the diagonals can be calculated using trigonometric relationships.

For the rhombus with an angle of ( \frac{3\pi}{8} ), let ( d_1 ) and ( d_2 ) be the lengths of the diagonals. Using trigonometric relationships, we find:

( d_1 = 15 \times \frac{1}{\sin(\frac{3\pi}{8})} ) and ( d_2 = 15 \times \frac{1}{\sin(\frac{5\pi}{8})} ).

Similarly, for the rhombus with an angle of ( \frac{11\pi}{12} ), let ( d_1' ) and ( d_2' ) be the lengths of the diagonals. Again using trigonometric relationships:

( d_1' = 15 \times \frac{1}{\sin(\frac{11\pi}{12})} ) and ( d_2' = 15 \times \frac{1}{\sin(\frac{\pi}{12})} ).

Calculate the areas of both rhombuses using the area formula, then find the difference between their areas.

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