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 `4`, its base's sides have lengths of `3`, and its base has a corner with an angle of `(5 pi)/6`. What is the pyramid's surface area?

Answer 1

T S A = 30.132

Explanation:

AB = BC = CD = DA = a = 3
Height OE = h = 4
OF = a/2 = 3/2 = 1.5
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(4^2+1.5^2) = color(red)(4.272)`

Area of `DCE = (1/2)*a*EF = (1/2)*3*4.272 = color(red)(6.408)`
Lateral surface area `= 4*Delta DCE = 4*6.408 = color(blue)(25.632)`

`/_C = pi - (5pi)/6 = (pi)/6`
Area of base ABCD `= a* a * sin /_C = 3^2 sin (pi/6) = 4.5`

T S A `= Lateral surface area + Base area`
T S A `=25.632 + 4.5 = color(purple)(30.132)`