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

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Answer 1 · 2017-12-06T12:50:25

T S A = 198.5344

AB = BC = CD = DA = a = 8
Height OE = h = 8
OF = a/2 = 8/2 = 4
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(8^2+4^2) = color(red)(8.9443)#

Area of #DCE = (1/2)*a*EF = (1/2)*8*8.9443 = color(red)(35.7772)#
Lateral surface area #= 4*Delta DCE = 4*35.7772 = color(blue)(143.1088)#

#/_C = pi/3, /_C/2 = pi/6#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=8sin(pi/6)= 4

#OC = d_1/2 = BC cos (C/2) = 8* cos (pi/6) = 6.9282

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*4) (2*6.9282) = color (blue)(55.4256)#

T S A #= Lateral surface area + Base area#
T S A # =143.1088 + 55.4256 = color(purple)(198.5344)#

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