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 #3 #, its base has sides of length #9 #, and its base has a corner with an angle of # pi/4 #. What is the pyramid's surface area?

1 Answer
Oct 18, 2017

Total Surface Area #= 200.2119#

Explanation:

AB = BC = CD = DA = a = 9
Height OE = h = 3
OF = a/2 = 9/2
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(3^2+(9/2)^2)=5.4083#

Area of #DCE = (1/2)*a*EF = (1/2)*9*5.4083 = 24.3375#
Lateral surface area #= 4*Delta DCE = 4*24.3375 = 94.3799#

#/_C = pi/4, /_C/2 = pi/8#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BC*sin (C/2)=9*sin(pi/8) = 6.364#
#OC = d_1/2 = BC cos (C/2) = 9* cos (pi/8) = 8.3149#

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*6.364)(2*8.3149) = 105.832#

Total Surface Area #= Lateral surface area + Base area. T S A # 94.3799 + 105.832 = 200.2119#
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