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

Answer 1

T S A = 85.4558

Explanation:

AB = BC = CD = DA = a = 6
Height OE = h = 4
OF = a/2 = 6/2 = 3
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(4^2+3^2) = color(red)(5)`

Area of `DCE = (1/2)*a*EF = (1/2)*6*5 = color(red)(15)`
Lateral surface area `= 4*Delta DCE = 4*15 = color(blue)(60)`

`/_C = pi - (3pi)/4 = (pi)/4`
Area of base ABCD `= a* a * sin /_C = 6^2 sin (pi/4) = 3.464`

T S A `= Lateral surface area + Base area`
T S A `=60 + 25.4558 = color(purple)(85.4558)`