A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #7 # and #8 # and the pyramid's height is #5 #. If one of the base's corners has an angle of #pi/4#, what is the pyramid's surface area?

Question

A pyramid has a parallelogram shaped base and a peak directly above its center.  Its base's sides have lengths of #7  # and #8  # and the pyramid's height is #5 #.  If one of the base's corners has an angle of #pi/4#, what is the pyramid's surface area?

Maverick Smith · Accepted Answer

total surface area pyramid #=28sqrt2+2sqrt(498)+7sqrt(33)#
total surface area pyramid #=124.4417455" "#square units

The intersection of the diagonals is where the parallelogram's center is located. Compute half-length of the longer diagonal #d_L# by cosine law for sides #d_L=sqrt(8^2+7^2-2*(7)*(8)*cos (135^@))# #d_L/2=1/2*sqrt(113+56sqrt(2))# As seen from the top of the pyramid, let x be one of the lengths of the edges of the triangular face that is directly on top of the longer diagonal. #x=sqrt((d_L/2)^2+h^2)# #x=sqrt((1/2*sqrt(113+56sqrt(2)))^2+5^2)# #x=8.54687018# Compute half-length of the shorter diagonal #d_S# by cosine law for sides #d_S=sqrt(8^2+7^2-2*(7)*(8)*cos (45^@))# #d_S/2=1/2*sqrt(113-56sqrt(2))# As seen from the top of the pyramid, let y be one of the lengths of the edges of the triangular face that is directly on top of the shorter diagonal. #y=sqrt((d_S/2)^2+h^2)# #y=sqrt((1/2*sqrt(113-56sqrt(2)))^2+5^2)# #y=5.783684823# Compute area #A_8# of triangular face with sides 8, x, and y, by Heron's Formula where #s_8=1/2*(8+x+y)=11.1652775# #A_8=sqrt(s_8*(s_8-8)*(s_8-x)*(s_8-y))# #A_8=sqrt(11.1652775*(11.1652775-8)*(11.1652775-8.54687018)(11.1652775-5.783684823))# #A_8=sqrt(498)# Compute area #A_7# of triangular face with sides 7, x, and y, by Heron's Formula where #s_7=1/2*(7+x+y)=10.6652775# #A_7=sqrt(s_7*(s_7-7)*(s_7-x)*(s_7-y))# #A_7=sqrt(10.6652775*(10.6652775-7)*(10.6652775-8.54687018)(10.6652775-5.783684823))# #A_7=(7*sqrt(33))/2# Compute Total Area #T_A:# #T_A=area base+2*A_8+2*A_7# #T_A=(7*sin 45^@)*(8)+2*sqrt(498)+2*(7*sqrt(33))/2# #T_A=28sqrt2+2sqrt(498)+7sqrt(33)# #T_A=124.4417455" "#square units May God bless you all. I hope this explanation helps.