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 `8`, and its base has a corner with an angle of `(7 pi)/8`. What is the pyramid's surface area?

Answer 1

T S A = 115.0021

Explanation:

AB = BC = CD = DA = a = 8
Height OE = h = 4
OF = a/2 = 8/2 = 4
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(4^2+4^2) = color(red)(5.6569)`

Area of `DCE = (1/2)*a*EF = (1/2)*8*5.6569 = color(red)(22.6276)`
Lateral surface area `= 4*Delta DCE = 4*22.6276 = color(blue)(90.5104)`

`/_C = (pi) - ((7pi)/8) = (pi)/8`
Area of base ABCD `= a* a * sin /_C = 8^2 sin (pi/8) = 24.4917`

T S A `= Lateral surface area + Base area`
T S A `=90.5104 + 24.4917 = color(purple)(115.0021)`