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 $8$ $8$ , its base has sides of length $6$ $6$ , and its base has a corner with an angle of $\frac{3 π}{4}$ $(3 pi)/4$ . What is the pyramid's surface area?

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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 $8$ $8$ , its base has sides of length $6$ $6$ , and its base has a corner with an angle of $\frac{3 π}{4}$ $(3 pi)/4$ . What is the pyramid's surface area?

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Answer 1 · 2017-10-11T18:23:33

Surface area of pyramid with rhombus base = 127.9839

Explanation:

Opposite base angles of the rhombus are $\frac{3 π}{4} & \frac{π}{4}$ $(3pi)/4 & pi/4$ respy.
$d 1 & d 2$ $d1 & d2$ be the two diagonals intersecting at right angle.
$sin (\frac{θ}{2}) = sin (\frac{\frac{π}{4}}{2}) = sin (\frac{π}{8}) = (\frac{d \frac{1}{2}}{6})$ $sin (theta/2) = sin ((pi/4)/2) = sin (pi/8)=((d1/2)/6)$
$sin (\frac{π}{8}) = d \frac{1}{12} or d 1 = 12 \cdot sin (\frac{π}{8}) =$ $sin(pi/8)=d1/12 or d1=12*sin(pi/8)=$ 4.5922

Similarly, $cos (\frac{π}{8}) = (\frac{d \frac{2}{6}}{2}) = (d \frac{2}{12})$ $cos(pi/8)=((d2/6)/2)=(d2/12)$
$d 2 = 12 \cdot cos (\frac{π}{8}) =$ $d2=12*cos(pi/8)=$ 11.0866

Area of Rhombus base $= \frac{d 1 \cdot d 2}{2} = \frac{4.5922 \cdot 11.0866}{2} =$ $=(d1*d2)/2=(4.5922*11.0866)/2=$ 25.4559

Height of one side surface $h 1 = \sqrt{{(\frac{b}{2})}^{2} + h^{2}}$ $h1=sqrt((b/2)^2+h^2)$
As $\frac{b}{2} = \frac{6}{2} = 3, h 1 = \sqrt{3^{2} + 8^{2}} = 8.544$ $b/2=6/2=3, h1=sqrt(3^2+8^2)=8.544$

Area of 4 sides of pyramid $= 4 \cdot (\frac{1}{2}) b \cdot h 1 = 2 \cdot 6 \cdot 8.544 =$ $=4*(1/2)b*h1=2*6*8.544=$ 102.588

Surface area of pyramid = base area + 4 side areas
#=25.4559+102.588=127.9839
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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 $8$ $8$ , its base has sides of length $6$ $6$ , and its base has a corner with an angle of $\frac{3 π}{4}$ $(3 pi)/4$ . What is the pyramid's surface area?

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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 $8$ $8$ , its base has sides of length $6$ $6$ , and its base has a corner with an angle of $\frac{3 π}{4}$ $(3 pi)/4$ . What is the pyramid's surface area?