A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #2 # and #12 # and the pyramid's height is #3 #. If one of the base's corners has an angle of #pi/4#, what is the pyramid's surface area?

1 Answer
Jan 7, 2018

Total Surface Area #= A_(base) + A_L = color(red)(68.3344)#

Explanation:

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Area of base #A_(base) = DC * AD * sin D = 12 * 2 * sin (pi/4)#

#A_(base) = 16.9706#

Area of #Delta ADE = (1/2) * AD * sqrt(EM^2 + ((DC)/2)^2)#

#A_(ADE) = (1/2) * 2 * sqrt(3^2 + ((12)/2)^2) = 6.7082#

#A_(ABE) = (1/2) * AB * sqrt(AB^2+ ((AD)/2)^2)#

#A_(ABE) = (1/2) * 12 * sqrt(3^2 + (2/2)^2) = 18.9737#

Lateral Surface Area #A_L = (2 * A_(ADE)) + (2 * A_(ABE))#

#A_L= (2*6.7082) + (2 * 18.9737) = 51.3638#

Total Surface Area
#A_T = A_(base) + A_L = 16.9706 + 51.3638 = color(red)68.3344#