A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is #4 #, its base has sides of length #6 #, and its base has a corner with an angle of #(3 pi)/4 #. What is the pyramid's surface area?

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Answer 1 · 2017-12-12T13:22:05

T S A = 85.4558

AB = BC = CD = DA = a = 6
Height OE = h = 4
OF = a/2 = 6/2 = 3
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(4^2+3^2) = color(red)(5)#

Area of #DCE = (1/2)*a*EF = (1/2)*6*5 = color(red)(15)#
Lateral surface area #= 4*Delta DCE = 4*15 = color(blue)(60)#

#/_C = pi - (3pi)/4 = (pi)/4#
Area of base ABCD #= a* a * sin /_C = 6^2 sin (pi/4) = 3.464#

T S A #= Lateral surface area + Base area#
T S A # =60 + 25.4558 = color(purple)(85.4558)#

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