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 $7$ $7$ , its base's sides have lengths of $5$ $5$ , and its base has a corner with an angle of $\frac{5 π}{8}$ $(5 pi)/8$ . 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 $7$ $7$ , its base's sides have lengths of $5$ $5$ , and its base has a corner with an angle of $\frac{5 π}{8}$ $(5 pi)/8$ . What is the pyramid's surface area?

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Answer 1 · 2018-01-23T09:40:12

Total Surface Area

$T S A = A_{T} = A_{L} + A_{R} = 23.097 + 74.3303 = 97.4273$ $color(green)( T S A = A_T = A_L + A_R = 23.097 + 74.3303 = 97.4273)$

Explanation:

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Area of Rhombus base $A_{R} = a \cdot a sin θ$ $A_R = a * a sin theta$

$A_{R} = 5 \cdot 5 \cdot sin (\frac{5 π}{8}) \approx 23.097$ $A_R = 5 * 5 * sin ((5pi)/8) ~~ 23.097$

Area of slant triangle $A_{t} = (\frac{1}{2}) a \cdot h_{l}$ $A_t = (1/2) a * h_l$ where $a$ $color(red)(a)$ is the rhombus base length and $h_{l}$ $color(red)(h_l)$ is the slant height of the triangle.

$h_{l} = \sqrt{h^{2} + {(\frac{a}{2})}^{2}}$ $h_l = sqrt(h^2 + (a/2)^2)$ where $h$ $color(red)(h)$ is the pyramid’s height.

Lateral Surface Area $A_{L} = 4 \cdot A_{t} = 4 \cdot (\frac{1}{2}) \cdot 5 \cdot \sqrt{7^{2} + {(\frac{5}{2})}^{2}}$ $A_L = 4 * A_t = 4 * (1/2) * 5 * sqrt(7^2 + (5/2)^2)$

$L S A = A_{L} = 74.3303$ $L S A = A_L = 74.3303$

Total Surface Area $T S A = A_{T} = A_{L} + A_{R} = 23.097 + 74.3303 = 97.4273$ $color(green)( T S A = A_T = A_L + A_R = 23.097 + 74.3303 = 97.4273)$

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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 $7$ $7$ , its base's sides have lengths of $5$ $5$ , and its base has a corner with an angle of $\frac{5 π}{8}$ $(5 pi)/8$ . What is the pyramid's surface area?

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1 Answer

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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 $7$ $7$ , its base's sides have lengths of $5$ $5$ , and its base has a corner with an angle of $\frac{5 π}{8}$ $(5 pi)/8$ . What is the pyramid's surface area?