Two rhombuses have sides with lengths of #9 #. If one rhombus has a corner with an angle of #(11pi)/12 # and the other has a corner with an angle of #(pi)/4 #, what is the difference between the areas of the rhombuses?

Question

Two rhombuses have sides with lengths of #9 #.  If one rhombus has a corner with an angle of #(11pi)/12  # and the other has a corner with an angle of #(pi)/4  #, what is the difference between the areas of the rhombuses?

Theodore Abad · Accepted Answer

#36.312units^2# to the nearest 3 decimal places

#:.(11cancelpi^1)/cancel12^1xxcancel180^15/cancelpi^1=165^@# #:.(cancelpi^1)/cancel4^1xxcancel180^45/cancelpi^1=45^@# :.Area of rhombus#=s^2xxsina# #:.9^2xxsin165^@=20.964units^2# #:.9^2xxsin45^@=57.276units^2# :.difference between 2 rhombusses # =57.276-20.964=36.312units^2#

Savannah Aguilar · 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.

For a rhombus with side length $ s $ and an interior angle $ \theta $, the diagonals are $ d_1 = s $ and $ d_2 = 2s \sin(\frac{\theta}{2}) $.

Given that the side length $ s = 9 $, and the angles are $ \frac{11\pi}{12} $ and $ \frac{\pi}{4} $, the diagonals for the first rhombus are $ d_1 = 9 $ and $ d_2 = 2 \times 9 \times \sin(\frac{11\pi}{24}) $, and for the second rhombus, $ d_1 = 9 $ and $ d_2 = 2 \times 9 \times \sin(\frac{\pi}{8}) $.

Calculate the areas of both rhombuses using the formula $ \text{Area} = \frac{d_1 \times d_2}{2} $ and find the difference between them.