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 `8`, its base has sides of length `2`, and its base has a corner with an angle of `(3 pi)/8`. What is the pyramid's surface area?

Answer 1

T S A = 35.9447

Explanation:

AB = BC = CD = DA = a = 2
Height OE = h = 8
OF = a/2 = 2/2 = 1
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(8^2+1^2) = color(red)(8.0623)`

Area of `DCE = (1/2)*a*EF = (1/2)*2*8.0623 = color(red)(8.0623)`
Lateral surface area `= 4*Delta DCE = 4*8.0623 = color(blue)(32.2492)`

`/_C = (3pi)/8`
Area of base ABCD `= a* a * sin /_C = 2^2 * sin (3pi)/8 = 3.6955`

T S A `= Lateral surface area + Base area`
T S A `=32.2492 + 3.6955 = color(purple)(35.9447)`