Two opposite sides of a parallelogram each have a length of #8 #. If one corner of the parallelogram has an angle of #( pi)/3 # and the parallelogram's area is #80 #, how long are the other two sides?

Solving for:
Length #b#

Given:
Area #A=80#
Length #a=8#
Angle #angleC=pi/3#

The area of a parallelogram is equal to the product #atimesh#, where #h# is the height (distance) between the #a#-sides.

#"        "A=ah#
#=>80=8h#
#=>10=h "      "therefore# the height is 10.

The side we're solving for, #b#, can be considered the hypotenuse of a right triangle with one angle #pi/3# and its opposite side with length #10#.

#"      "sin" "angle" "C""="opp"/"hyp"#

#=>sin(pi/3)=10/b#

#=>"   "sqrt2/2"    "=10/b#

#=>"       "b"     "=(2*10)/sqrt2color(blue)(*sqrt2/sqrt2)#        (rationalize)

#=>"       "b"     "=(20sqrt2)/2=10sqrt2#

#therefore# the length #b# is #10sqrt2#.

Bonus:

From this, we can actually get a general formula that relates a parallelogram's area to its side lengths and the one angle:

Since #sinangleC=h/b#, we can solve this for #h# to get
#h=bsinangleC#

And since we know #A=ah#, we can plug the above RHS in for #h#:

#"       "A=ah#
#=>A=ab sinangleC#

You may have seen a similar formula for the area of a triangle:
#A_triangle=1/2ab sinC#
And we know a parallelogram can be geometrically split into two identical triangles, thus, twice the area of such a triangle is the area of the whole parallelogram.