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

1 Answer
Dec 2, 2017

T S A #= color(purple)(190.8252)#

Explanation:

AB = BC = CD = DA = a = 8
Height OE = h = 3
OF = a/2 = 8/2 = 4
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(3^2+4^2) = color(red)5#

Area of #DCE = (1/2)*a*EF = (1/2)*8*5 = color(red)(20)#
Lateral surface area #= 4*Delta DCE = 4*20 = color(blue)(80)#

#/_C = pi/3, /_C/2 = pi/6#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BC*sin (C/2)=8*sin(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*8) (2*6.9282) = color (blue)(110.8512)#

Total Surface Area #= Lateral surface area + Base area#
T S A # =80 + 110.8252 = color(purple)(190.8252)#

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