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 #3 #, its base has sides of length #5 #, and its base has a corner with an angle of # pi/4 #. What is the pyramid's surface area?

1 Answer
Feb 4, 2018

#color(green)(T S A)# #color(green)(A_T = A_L + A_R = 74.4063)#

Explanation:

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Total Surface Area of the pyramid is sum of the areas of the rhombus base #(A_R)# and the four side triangles #(A_L= 4 * A_t)#
#A_T = A_L + A_R = 4A_t + A_R#

#A_R = a*asin theta = 5 * 5 sin (pi/4) = 25sqrt2 = 35.3553##

A_L = 4 A_t = 4 * (1/2) (a * l)# where l is the slant height of the triangle.

But # l = sqrt((a/2)^2 + h^2) = sqrt((5/2)^2 + 3^2) = 3.9051#

#:. A_L = 4 * (1/2) * 5 * 3.9051 = 39.051#

#color(green)(T S A)# #color(green)(A_T = A_L + A_R = 39.051 + 35.3553 = 74.4063)#