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

Answer 1

T S A = 170.0112

Explanation:

AB = BC = CD = DA = a = 8
Height OE = h = 6
OF = a/2 = 1/2 = 4
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(6^2+4^2) = color(red)(7.2111)`

Area of `DCE = (1/2)*a*EF = (1/2)*8*7.2111 = color(red)(28.8444)`
Lateral surface area `= 4*Delta DCE = 4*28.8444 = color(blue)(115.3776`)#

`/_C = pi/3, /_C/2 = pi/6`
diagonal `AC = d_1` & diagonal `BD = d_2`
#OB = d_2/2 = BCsin (C/2)=8sin(pi/6)= 4

#OC = d_1/2 = BC cos (C/2) = 8* cos (pi/6) = 6.9282

Area of base ABCD `= (1/2)*d_1*d_2 = (1/2)(2*4) (2*6.9282) = color (blue)(54.6336)`

T S A `= Lateral surface area + Base area`
T S A `=115.3776 + 54.6336 = color(purple)(170.0112)`