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 `2`, 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 = 31.7485

Explanation:

AB = BC = CD = DA = a = 2
Height OE = h = 7
OF = a/2 = 2/2 = 1
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(7^2+1^2) = color(red)(7.0711)`

Area of `DCE = (1/2)*a*EF = (1/2)*2*7.0711 = color(red)(7.0711)`
Lateral surface area `= 4*Delta DCE = 4*7.0711 = color(blue)(28.2844)`

`/_C = pi - (2pi)/3 = (pi)/3`
Area of base ABCD `= a* a * sin /_C = 2^2 sin (pi/3) = 3.4641`

T S A `= Lateral surface area + Base area`
T S A `=28.2844 + 3.4641 = color(purple)(31.7485)`