A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #4 # and #7 # and the pyramid's height is #6 #. 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 #4  # and #7  # and the pyramid's height is #6 #.  If one of the base's corners has an angle of #pi/4  #, what is the pyramid's surface area?

Allison Allen · Accepted Answer

T S A = 91.8558

#CH = 4 * sin ((pi)/4) = 2.8284#
Area of parallelogram base #= a * b1 = 7*2.8284 = color(red)(19.7988)#

#EF = h_1 = sqrt(h^2 + (a/2)^2) = sqrt(6^2+ (7/2)^2)= 6.9462#
Area of # Delta AED = BEC = (1/2)*b*h_1 = (1/2)*4* 6.9462= #color(red)(13.8924)#

#EG = h_2 = sqrt(h^2+(b/2)^2 ) = sqrt(6^2+(4/2)^2 )= 6.3246#
Area of #Delta = CED = AEC = (1/2)*a*h_2 = (1/2)*7*6.3246 = color(red)( 22.1361)#

Lateral surface area = #2* DeltaAED + 2*Delta CED#
#=( 2 * 13.8924)+ (2* 22.1361) = color(red)(72.057)#

Total surface area =Area of parallelogram base + Lateral surface area # = 19.7988 + 72.057 = 91.8558#

Wyatt Anderson · Answer

undefined The surface area of the pyramid can be calculated using the formula:

$$ A = 2B + P $$

Where $ B $ is the area of the base and $ P $ is the lateral surface area.

Given that the base is a parallelogram, the area $ B $ can be calculated as the product of the base length and the corresponding height.

$$ B = \text{base length} \times \text{height} $$

For a parallelogram, the height can be found by dropping a perpendicular from one of the vertices to the base.

$$ \text{height} = 4 \times \sin\left(\frac{\pi}{4}\right) $$

The lateral surface area $ P $ can be found by adding the areas of the four triangular faces.

$$ P = 4 \times \frac{1}{2} \times \text{base length} \times \text{slant height} $$

The slant height can be found using the Pythagorean theorem:

$$ \text{slant height} = \sqrt{\text{height}^2 + \left(\frac{1}{2} \times \text{base length}\right)^2} $$

Now, substitute the given values into the formulas and compute the surface area.