Two rhombuses have sides with lengths of #4 #. If one rhombus has a corner with an angle of #(11pi)/12 # and the other has a corner with an angle of #(3pi)/8 #, what is the difference between the areas of the rhombuses?

Question

Two rhombuses have sides with lengths of #4 #.  If one rhombus has a corner with an angle of #(11pi)/12  # and the other has a corner with an angle of #(3pi)/8  #, what is the difference between the areas of the rhombuses?

Mason Adams · Accepted Answer

Difference in areas between the two rhombuses is 10.6408

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.

Maverick Smith · Answer

undefined The area of a rhombus can be calculated using the formula:

$$ \text{Area} = \frac{{d_1 \times d_2}}{2} $$

where $ d_1 $ and $ d_2 $ are the lengths of the diagonals.

In this case, since the length of each side of the rhombuses is 4, the lengths of their diagonals can be found using trigonometry.

For each rhombus:
$$ d_1 = 4 \times \sin(\text{angle}) $$
$$ d_2 = 4 \times \sin(\frac{\pi}{2} - \text{angle}) $$

Calculate the diagonals for both rhombuses using the given angles, then find the areas using the formula mentioned above. Finally, find the difference between the areas of the two rhombuses.