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 `3`, its base's sides have lengths of `2`, and its base has a corner with an angle of `(7 pi)/8`. What is the pyramid's surface area?

Answer 1

Total Surface Area of the pyramid = `color(purple)(14.1799)`

Explanation:

AB = BC = CD = DA = a = 2
Height OE = h = 3
OF = a/2 = 2/2 = 1
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(3^2+1^2) = color(red)3.1623`

Area of `DCE = (1/2)*a*EF = (1/2)*2*3.1623= color(red)(3.1623)`
Lateral surface area `= 4*Delta DCE = 4*3.1623= color(blue)(12.6491)`

`/_C = pi/8, /_C/2 = pi/16`
diagonal `AC = d_1` & diagonal `BD = d_2`
`OB = d_2/2 = BC*sin (C/2)=2*sin(pi/16)= **0.3902**`

#OC = d_1/2 = BC cos (C/2) = 2* cos (pi/16) = 1.9616

Area of base ABCD `= (1/2)*d_1*d_2 = (1/2)(2*0.3902)(2*1.9616) = color (blue)(1.5308)`

Total Surface Area `= Lateral surface area + Base area`
T S A `=12.6491 + 1.5308 = color(purple)(14.1799)`