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)/4 #, 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)/4  #, what is the difference between the areas of the rhombuses?

Evelyn Arboleda · Accepted Answer

Difference in areas between the two rhombuses is 7.1724

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.

Evelyn Arboleda · Answer

undefined To find the difference between the areas of the rhombuses, we first need to calculate the areas of both rhombuses.

The area $ A $ of a rhombus can be calculated using the formula $ A = \frac{1}{2} \times d_1 \times d_2 $, where $ d_1 $ and $ d_2 $ are the lengths of the diagonals.

For a rhombus with side length $ s $ and an angle $ \theta $ at one of its corners, the diagonals are given by $ d_1 = s $ and $ d_2 = s \cdot \sqrt{2 - 2\cos(\theta)} $.

Given that both rhombuses have side lengths of 4, and using the angles provided:

For the first rhombus with angle $ \frac{11\pi}{12} $:
$$ d_1 = 4 $$
$$ d_2 = 4 \cdot \sqrt{2 - 2\cos\left(\frac{11\pi}{12}\right)} $$

For the second rhombus with angle $ \frac{3\pi}{4} $:
$$ d_1 = 4 $$
$$ d_2 = 4 \cdot \sqrt{2 - 2\cos\left(\frac{3\pi}{4}\right)} $$

Now, calculate the areas of both rhombuses using the formula $ A = \frac{1}{2} \times d_1 \times d_2 $ for each.

Finally, subtract the smaller area from the larger area to find the difference between the areas of the rhombuses.