A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #3 # and #1 # 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 #3  # and #1  # 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?

Noah Atkinson · Accepted Answer

T S A is 22.4158

#CH = 1 * sin (pi/4) = 0.707#
Area of parallelogram base #= a * b1 = 3*0.707 = color(red)(2.121 )#

#EF = h_1 = sqrt(h^2 + (a/2)^2) = sqrt(5^2+ (3/2)^2)= 5.22#
Area of # Delta AED = BEC = (1/2)*b*h_1 = (1/2)*1* 5.22= #color(red)(2.61)#

#EG = h_2 = sqrt(h^2+(b/2)^2 ) = sqrt(5^2+(1/2)^2 )= 5.0249#
Area of #Delta = CED = AEC = (1/2)*a*h_2 = (1/2)*3*5.0249 = color(red)( 7.5374)#

Lateral surface area = #2* DeltaAED + 2*Delta CED#
#=( 2 * 2.61)+ (2* 7.5374) = color(red)(20.2948)#

Total surface area =Area of parallelogram base + Lateral surface area # = 2.121 + 20.2948 = 22.4158#

Caroline Aucoin · 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:
   Since the base of the pyramid is a parallelogram, its area can be calculated as the product of the length of one side and the corresponding height perpendicular to that side.
   Given the base sides are 3 and 1, and the height is 5, we can find the area of the base using the formula for the area of a parallelogram: $A_{\text{base}} = \text{base} \times \text{height}$.

2. Area of the Triangular Faces:
   Each triangular face of the pyramid is an isosceles triangle with two sides equal to the slant height of the pyramid and one side equal to the length of a side of the base. The slant height can be found using the Pythagorean theorem, knowing the height and half of the base's diagonal.
   Once the slant height is found, the area of each triangular face can be calculated using the formula for the area of a triangle: $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}$.

3. Total Surface Area:
   Add the area of the base and the combined areas of the four triangular faces to find the total surface area of the pyramid.

Performing the calculations based on the given dimensions and angles, we can find the surface area of the pyramid.