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

Answer 1

Total Surface Area #T S A = 101.4925

Explanation:

AB = BC = CD = DA = a = 5
Height OE = h = 8
OF = a/2 = 1/2 = 2.5
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(8^2+2.5^2) = color(red)(8.3815)`

Area of `DCE = (1/2)*a*EF = (1/2)*5*8.3815 = color(red)(20.9538)`
Lateral surface area `= 4*Delta DCE = 4*20.9538 = color(blue)(83.815`)#

`/_C = (3pi)/4, /_C/2 = (3pi)/8`
diagonal `AC = d_1` & diagonal `BD = d_2`
#OB = d_2/2 = BCsin (C/2)=5sin((3pi)/8)= 4.6194

#OC = d_1/2 = BC cos (C/2) = 5 cos ((3pi)/8) = 1.9134*

Area of base ABCD `= (1/2)*d_1*d_2 = (1/2)(2*4.6194) (2*1.9134) = color (blue)(17.6775)`

T S A `= Lateral surface area + Base area`
T S A `=83.815 + 17.6775 = color(purple)(101.4925)`