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

1 Answer
Dec 3, 2017

T S A #= color(purple)(95.98)#

Explanation:

AB = BC = CD = DA = a = 5
Height OE = h = 7
OF = a/2 = 5/2 = 2.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(7^2+2.5^2) = color(red)(7.433)#

Area of #DCE = (1/2)*a*EF = (1/2)*5*7.433 = color(red)(18.5825)#
Lateral surface area #= 4*Delta DCE = 4*18.5825 = color(blue)(74.33)#

#/_C = pi/3, /_C/2 = pi/6#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=5sin(pi/6)= 2.5

#OC = d_1/2 = BC cos (C/2) = 5* cos (pi/6) = 4.33

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*2.5) (2*4.33) = color (blue)(21.65)#

T S A #= Lateral surface area + Base area#
T S A # =74.33 + 21.65 = color(purple)(95.98)#

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