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

Answer 1

T S A = 33.9409

Explanation:

AB = BC = CD = DA = a = 4
Height OE = h = 2
OF = a/2 = 4/2 = 2
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(2^2+2^2) = color(red)(2.8284)`

Area of `DCE = (1/2)*a*EF = (1/2)*4*2.8284 = color(red)(5.6568)`
Lateral surface area `= 4*Delta DCE = 4*5.6568 = color(blue)(22.6272)`

`/_C = (pi)/4`
Area of base ABCD `= a* a * sin /_C = 4^2 sin (pi/4) = 11.3137`

T S A `= Lateral surface area + Base area`
T S A `=22.6272 + 11.3137 = color(purple)(33.9409)`