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 #(3pi)/8 #, what is the difference between the areas of the rhombuses?

1 Answer
Oct 20, 2017

Difference in areas between the two rhombus = 4.1618

Explanation:

Area of a rhombus = #(d_1 * d_2) / 2#
Where #d_1 , d_2 # are diagonals.
#side = a#

Diagonals bisect each other at right angles.

#:. d_1/2 = (a/2 )* sin (theta /2)#
#d_2 /2 = (a/2) * cos (theta/2)#

Rhombus #1 : side= 5 and /_theta = pi/12#
#d_1 = 5 * sin (pi/24) = 0.6526#
#d_2 = 5 * cos (pi/24) = 4.9752#
#Area = (d_1 * d_2) / 2 = (0.6526 * 4.9752)/2 = 1.6125#

Rhombus #1 : side= 5 and /_theta = (3pi)/8#

Rhombus #1 : side= 5 and /_theta = pi/12#
#d_1 = 5 * sin ((3pi)/16) = 4.1573#
#d_2 = 5 * cos ((3pi)/16)= 2.7779#
#Area = (d_1 * d_2) / 2 = (4.1573 * 2.7779)/2 = 5.7743#

Diff. in areas between the Rhombus = 5.7743 - 1.6125 = 4.1618#