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?

Question

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?

Sadie Adams · Accepted Answer

Difference between areas of the two rhombuses is 44.3763

Area of rhombus #= (1/2) * d_1 * d_2 or a * h#
Where #d_1 , d_2 # are the diagonals, a is the side and h is the altitude.

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

Carson Arnold · Answer

undefined 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.