Two rhombuses have sides with lengths of #5 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(7pi)/12 #, what is the difference between the areas of the rhombuses?

Question

Two rhombuses have sides with lengths of #5 #.  If one rhombus has a corner with an angle of #pi/12  # and the other has a corner with an angle of #(7pi)/12  #, what is the difference between the areas of the rhombuses?

Wyatt Anderson · Accepted Answer

#=17.68# Area of the rhombus with angle #theta=(pi)/12# and Side #a=5# is #=a^2 sin theta# #=5^2 sin((pi)/12)# #=25(0.2588)# #=6.47# Area of the rhombus with angle #theta=(7pi)/12# and Side #a=5# is #=5^2 sin theta# #=5^2 sin((7pi)/12)# #=25(0.966)# #=24.15# So difference in Area#=24.15-6.47=17.68#

Sadie Adams · Answer

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

The formula for the area of a rhombus is given by: 
Area = (diagonal1 * diagonal2) / 2.

Since the diagonals of a rhombus bisect each other at right angles, each diagonal forms two congruent right triangles. Thus, we can use trigonometry to find the length of each diagonal.

Given that the side length of each rhombus is 5 units and the angle at one corner of the first rhombus is π/12, we can use trigonometric ratios to find the length of the diagonals.

For the first rhombus:
Using sine, sin(π/12) = (diagonal / 5), 
Solving for diagonal: diagonal1 = 5 * sin(π/12).

For the second rhombus:
Since the angle at one corner is (7π)/12, we use the same trigonometric approach:
sin((7π)/12) = (diagonal / 5), 
Solving for diagonal: diagonal2 = 5 * sin((7π)/12).

Once we have the lengths of the diagonals for both rhombuses, we can plug these values into the area formula to find the area of each rhombus.

Finally, we subtract the area of the second rhombus from the area of the first rhombus to find the difference in their areas.

Difference = Area of first rhombus - Area of second rhombus.