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 #5 #, and its base has a corner with an angle of #(5 pi)/8 #. What is the pyramid's surface area?

1 Answer
Jan 23, 2018

Total Surface Area

#color(green)( T S A = A_T = A_L + A_R = 23.097 + 74.3303 = 97.4273)#

Explanation:

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Area of Rhombus base #A_R = a * a sin theta#

#A_R = 5 * 5 * sin ((5pi)/8) ~~ 23.097#

Area of slant triangle #A_t = (1/2) a * h_l# where #color(red)(a)# is the rhombus base length and #color(red)(h_l)#is the slant height of the triangle.

#h_l = sqrt(h^2 + (a/2)^2)# where #color(red)(h)# is the pyramid’s height.

Lateral Surface Area #A_L = 4 * A_t = 4 * (1/2) * 5 * sqrt(7^2 + (5/2)^2)#

#L S A = A_L = 74.3303#

Total Surface Area #color(green)( T S A = A_T = A_L + A_R = 23.097 + 74.3303 = 97.4273)#