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

1 Answer
sankarankalyanam
Dec 25, 2017

T S A = 30.132

Explanation:

AB = BC = CD = DA = a = 3
Height OE = h = 4
OF = a/2 = 3/2 = 1.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(4^2+1.5^2) = color(red)(4.272)#

Area of #DCE = (1/2)*a*EF = (1/2)*3*4.272 = color(red)(6.408)#
Lateral surface area #= 4*Delta DCE = 4*6.408 = color(blue)(25.632)#

#/_C = pi - (5pi)/6 = (pi)/6#
Area of base ABCD #= a* a * sin /_C = 3^2 sin (pi/6) = 4.5#

T S A #= Lateral surface area + Base area#
T S A # =25.632 + 4.5 = color(purple)(30.132)#

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