Two rhombuses have sides with lengths of #12 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(5pi)/8 #, what is the difference between the areas of the rhombuses?

Question

Two rhombuses have sides with lengths of #12 #.  If one rhombus has a corner with an angle of #pi/12  # and the other has a corner with an angle of #(5pi)/8  #, what is the difference between the areas of the rhombuses?

Mason Adams · Accepted Answer

Difference in areas between the two rhombuses is 95.7696

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.

Violet Aldrich · Answer

undefined To find the difference between the areas of the two rhombuses, we need to calculate the areas of each rhombus and then find the difference.

The formula for the area of a rhombus is given by $ \text{Area} = \frac{d_1 \times d_2}{2} $, where $d_1$ and $d_2$ are the lengths of the diagonals of the rhombus.

Given that the sides of both rhombuses have lengths of 12, we can use trigonometric properties to find the lengths of the diagonals.

For the rhombus with a corner angle of $ \frac{\pi}{12} $:
- Since the opposite angles of a rhombus are congruent, the angle between the diagonals is $ \frac{\pi}{12} + \frac{\pi}{12} = \frac{\pi}{6} $.
- Using trigonometric properties, we can find that $ d_1 = 12 \times \csc(\frac{\pi}{6}) $ and $ d_2 = 12 \times \csc(\frac{\pi}{6}) $.

For the rhombus with a corner angle of $ \frac{5\pi}{8} $:
- Similarly, the angle between the diagonals is $ \frac{5\pi}{8} + \frac{5\pi}{8} = \frac{5\pi}{4} $.
- Therefore, $ d_1 = 12 \times \csc(\frac{5\pi}{4}) $ and $ d_2 = 12 \times \csc(\frac{5\pi}{4}) $.

Now, calculate the areas of both rhombuses using the formula $ \text{Area} = \frac{d_1 \times d_2}{2} $, and then find the difference between the two areas.