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 `1`, 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 = 14.9016

Explanation:

AB = BC = CD = DA = a = 1
Height OE = h = 7
OF = a/2 = 1/2 = 0.5
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(7^2+0.5^2) = color(red)(7.0178)`

Area of `DCE = (1/2)*a*EF = (1/2)*1*7.0178 = color(red)(3.5089)`
Lateral surface area `= 4*Delta DCE = 4*3.5089 = color(blue)(14.0356)`

`/_C = pi - (2pi)/3 = (pi)/3`
Area of base ABCD `= a* a * sin /_C = 1^2 sin (pi/3) = 0.866`

T S A `= Lateral surface area + Base area`
T S A `=14.0356 + 0.866 = color(purple)(14.9016)`