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 #4 #, its base's sides have lengths of #1 #, and its base has a corner with an angle of #(5 pi)/6 #. What is the pyramid's surface area?

1 Answer
Dec 12, 2017

T S A = 8.5624

Explanation:

AB = BC = CD = DA = a = 1
Height OE = h = 4
OF = a/2 = 1/2 = 0.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(4^2+0.5^2) = color(red)(4.0311)#

Area of #DCE = (1/2)*a*EF = (1/2)*1*4.0311 = color(red)(2.0156)#
Lateral surface area #= 4*Delta DCE = 4*2.0156 = color(blue)(8.0624)#

#/_C =(pi) - ((5pi)/6) = (pi)/6#
Area of base ABCD #= a* a * sin /_C = 1^2 sin (pi/6) = 0.5#

T S A #= Lateral surface area + Base area#
T S A # =8.0624 + 0.5 = color(purple)(8.5624)#

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