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 #2 #, and its base has a corner with an angle of #(3 pi)/8 #. What is the pyramid's surface area?
1 Answer
A = surface area
Explanation:
The surface area of a pyramid is
#A = "base area" xx 1/2 xx "base perimeter" xx "slant height"#
The base is the rhombus:
Area of a rhombus
where
The internal angles of a quadrilateral
The diagonals of a rhombus are perpendicular.
The diagonals bisect the angles. This means if one corner angle is
Use trigonometry to find the diagonal lengths:
#BE = DE = 2 sin 3/16 pi = 1.11114# ;#BD = 2.22228#
#AE = CE = 2cos 3/16 pi = 1.66294# ;#AC = 3.32588#
The base area is
Base perimeter =
To find the slant height use Pythagorean Theorem.
slant height =
where
Slant height = sqrt (.9239^2 + 3^2) = 3.139.
The Pyramid's surface area is
#A = 3.6955 (1/2) (8)(3.139) = 46.4# #units^2#