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 #8 #, its base has sides of length #5 #, and its base has a corner with an angle of #(3 pi)/4 #. What is the pyramid's surface area?

1 Answer
Dec 6, 2017

Total Surface Area #T S A = 101.4925

Explanation:

AB = BC = CD = DA = a = 5
Height OE = h = 8
OF = a/2 = 1/2 = 2.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(8^2+2.5^2) = color(red)(8.3815)#

Area of #DCE = (1/2)*a*EF = (1/2)*5*8.3815 = color(red)(20.9538)#
Lateral surface area #= 4*Delta DCE = 4*20.9538 = color(blue)(83.815#)#

#/_C = (3pi)/4, /_C/2 = (3pi)/8#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=5sin((3pi)/8)= 4.6194

#OC = d_1/2 = BC cos (C/2) = 5* cos ((3pi)/8) = 1.9134

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*4.6194) (2*1.9134) = color (blue)(17.6775)#

T S A #= Lateral surface area + Base area#
T S A # =83.815 + 17.6775 = color(purple)(101.4925)#

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