Question

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`, its base has sides of length `9`, and its base has a corner with an angle of `(2 pi)/3`. What is the pyramid's surface area?

Answer 1

T S A = 219.9389

Explanation:

AB = BC = CD = DA = a = 9
Height OE = h = 7
OF = a/2 = 9/2 = 4.5
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(7^2+4.5^2) = color(red)(8.3217)`

Area of `DCE = (1/2)*a*EF = (1/2)*9*8.3217 = color(red)(37.4477)`
Lateral surface area `= 4*Delta DCE = 4*37.4477 = color(blue)(149.7908)`

`/_C = pi - (2pi)/3 = (pi)/3`
Area of base ABCD `= a* a * sin /_C = 9^2 * sin (pi/3) = 70.1481`

T S A `= Lateral surface area + Base area`
T S A `=149.7908 + 70.1481 = color(purple)(219.9389)`

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