Two rhombuses have sides with lengths of #4 #. If one rhombus has a corner with an angle of #(7pi)/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 #(7pi)/12  # and the other has a corner with an angle of #(3pi)/4  #, what is the difference between the areas of the rhombuses?

Savannah Aguilar · Accepted Answer

Area difference: #4.1411# (approx.)

where #theta# is an interior angle of the rhombus.

For rhombus #R1# with #theta=(7pi)/12# and #s=4#
#color(white)("XXX")"Area"_(R1) = 4 xx 4 xx sin((7pi)/12) ~~15.54481#

For rhombus #R2# with #theta=(3pi)/4# and #s=4#
#color(white)("XXX")"Area"_(R2)=4 xx 4 xx sin((3pi)/4) ~~11.31371#

The difference in the ares of the rhombi is
#color(white)("XXX")15.54481 - 11.31371 = 4.1411#

James Atkins · Answer

#4.14#

(Using trigonometry) We need to calculate the area of these rhombus' with given sides and an angle.And,we need to subtract their areas. Here comes a handy formula #color(blue)("Area of rhombus"=s^2sin(a)# Where, #color(orange)(s="side"and a="one of the angles"# So,we calculate the area of the first rhombus #rarrs^2sin(a)# #rarr4^2sin((7pi)/12)# #rarr16sin(105^circ)# #rarr16*0.96# #color(green)(rArr15.45# Now we calculate the area of the second rhombus #rarrs^2sin(a)# #rarr4^2sin((3pi)/4)# #rarr16sin(135^circ)# #rarr16*0.7# #color(green)(rArr11.31# Subtract the areas #rArr15.45-11.31=4.14# #:."The difference between the area are approximately"# #color(blue)(4.14# #("Note: these are approximations only")#

Aurora Ayala · Answer

undefined The difference between the areas of the rhombuses is 4 square units.