# A parallelogram has sides A, B, C, and D. Sides A and B have a length of #7 # and sides C and D have a length of # 4 #. If the angle between sides A and C is #(7 pi)/12 #, what is the area of the parallelogram?

##### 1 Answer

The area is

#### Explanation:

Let the angle between sides

The formula to compute the area of the parallelogram is

#"Area" = A * C * sin(alpha)#

#= 7 * 4 * sin((7pi)/12)#

#= 28 sin((7pi)/12)#

So, the only thing left to do is compute

Let me show how to do this without the calculator but with some basic knowledge of

# sin((7pi)/12) = sin(pi/4 + pi/3)#

... use the formula

#= sin(pi/4) * cos(pi/3) + cos(pi/4) * sin(pi/3)#

#= 1/sqrt(2) * 1/2 + 1/sqrt(2) * sqrt(3)/2 #

#= (1 + sqrt(3))/(2sqrt(2)) #

#= (sqrt(2) + sqrt(6))/4 #

Thus, you have the area of

#"Area" = 28 sin((7pi)/12) = 28 * (sqrt(2) + sqrt(6))/4 = 7 * (sqrt(2) + sqrt(6)) ~~ 27.05 " units"^2#