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 $2$ $2$ , its base has sides of length $1$ $1$ , and its base has a corner with an angle of $\frac{2 π}{3}$ $(2 pi)/3$ . 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 $2$ $2$ , its base has sides of length $1$ $1$ , and its base has a corner with an angle of $\frac{2 π}{3}$ $(2 pi)/3$ . What is the pyramid's surface area?

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Answer 1 · 2018-01-27T06:31:54

Total Surface Area $T S A = 4.9892$ $color(brown)(T S A) color(brown)(= 4.9892)$

Explanation:

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Pyramid's Total Surface Area (T S A) $A_{T}$ $A_T$ =Lateral Surface Area + Rhombus Base Area $(A_{R})$ $(A_R)$

Lateral Surface Area (L S A)= 4 * Area of Slant Triangle (A_S)

i.e. $A_{T} = A_{B} + (4 \cdot A_{S})$ $A_T = A_B + (4 * A_S)$

To find $A_{B}, A_{S}$ $A_B, A_S$

Let a be the rhombus side, theta the corner angle and h the height of the pyramid.

Given : $a = 1, θ = \frac{2 π}{3}, h = 2$ $a = 1, theta = (2pi)/3, h = 2$

$A_{R} = a^{2} sin θ = 1^{2} sin (\frac{2 π}{3}) = \frac{\sqrt{3}}{2} = 0.866$ $color(red)(A_R) = a^2 sin theta = 1^2 sin ((2pi)/3) color(red)(= sqrt3 /2 = 0.866)$

$A_{S} = (\frac{1}{2}) a \cdot \sqrt{h^{2} + {(\frac{a}{2})}^{2}}$ $A_S = (1/2) a * sqrt(h^2 + (a/2)^2)$

$A_{S} = (\frac{1}{2}) (1) \cdot \sqrt{2^{2} + {(\frac{1}{2})}^{2}} = 1.0308$ $color(red)(A_S) = (1/2)(1) * sqrt(2^2 + (1/2)^2) color(red)(= 1.0308)$

$T S A = A_{T} = A_{R} + (4 \cdot A_{S}) = 0.866 + (4 \cdot 1.0308) = 4.9892$ $color(red)(T S A) = A_T = A_R + (4 * A_S) = 0.866 + (4 * 1.0308) color(red)(= 4.9892)$

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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 $2$ $2$ , its base has sides of length $1$ $1$ , and its base has a corner with an angle of $\frac{2 π}{3}$ $(2 pi)/3$ . What is the pyramid's surface area?

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Explanation:

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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 $2$ $2$ , its base has sides of length $1$ $1$ , and its base has a corner with an angle of $\frac{2 π}{3}$ $(2 pi)/3$ . What is the pyramid's surface area?