A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #7 # and #2 # and the pyramid's height is #8 #. If one of the base's corners has an angle of #(3pi)/8#, 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 #2  # and the pyramid's height is #8 #.  If one of the base's corners has an angle of #(3pi)/8#, what is the pyramid's surface area?

Mason Adams · Accepted Answer

T S A = 86.835

#CH = 2 * sin ((3pi)/8) = 1.8478#
Area of parallelogram base #= a * b1 = 7*1.8478 = color(red)(12.9346)#

#EF = h_1 = sqrt(h^2 + (a/2)^2) = sqrt(8^2+ (7/2)^2)= 8.7321#
Area of # Delta AED = BEC = (1/2)*b*h_1 = (1/2)*2* 8.7321= #color(red)(8.7321)#

#EG = h_2 = sqrt(h^2+(b/2)^2 ) = sqrt(8^2+(2/2)^2 )= 8.0623#
Area of #Delta = CED = AEC = (1/2)*a*h_2 = (1/2)*7*8.0623 = color(red)( 28.2181)#

Lateral surface area = #2* DeltaAED + 2*Delta CED#
#=( 2 * 8.7321)+ (2* 28.2181) = color(red)(73.643)#

Total surface area =Area of parallelogram base + Lateral surface area # = 12.9346 + 73.9004 = 86.835#

Allison Allen · Answer

undefined To find the surface area of the pyramid, we need to calculate the area of the base and the area of the four triangular faces.

1. Area of the base (parallelogram):
   Area = base × height
   Area = 7 × 2 = 14 square units

2. To find the area of each triangular face:
   We first need to find the length of the slant height (l) using the Pythagorean theorem:
   l^2 = h^2 + (1/2 * base)^2
   l^2 = 8^2 + (1/2 * 7)^2
   l^2 = 64 + 12.25
   l^2 = 76.25
   l ≈ √76.25 ≈ 8.72 units

Now, we can find the area of one triangular face using the formula:
   Area of one triangular face = (1/2) × base × height
   Area of one triangular face = (1/2) × 7 × 8 ≈ 28 square units

Since there are four triangular faces, the total area of the triangular faces is:
   Total area of triangular faces = 4 × 28 = 112 square units

3. Total surface area of the pyramid:
   Total surface area = area of base + area of triangular faces
   Total surface area = 14 + 112 = 126 square units

Therefore, the surface area of the pyramid is 126 square units.