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's sides have lengths of #1 #, and its base has a corner with an angle of #(7 pi)/8 #. What is the pyramid's surface area?

Question

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's sides have lengths of #1 #, and its base has a corner with an angle of #(7 pi)/8 #.  What is the pyramid's surface area?

Noah Atkinson · Accepted Answer

T S A = 14.4183

AB = BC = CD = DA = a = 1
Height OE = h = 7
OF = a/2 = 1/2 = 0.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(7^2+0.5^2) = color(red)(7.0178)#

Area of #DCE = (1/2)*a*EF = (1/2)*1*7.0178 = color(red)(3.5089)#
Lateral surface area #= 4*Delta DCE = 4*3.5089 = color(blue)(14.0356)#

#/_C = (7pi)/8, /_C/2 = (7pi)/16#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=1sin((7pi)/16)= 0.9808

#OC = d_1/2 = BC cos (C/2) = 1* cos ((7pi)/16) = 0.1951

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*0.1951) (2*0.9808) = color (blue)(0.3827)#

T S A #= Lateral surface area + Base area#
T S A # =14.0356 + 0.3827 = color(purple)(14.4183)#

Carson Arnold · Answer

undefined The surface area of the pyramid is approximately 13.07 square units.