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 `5`, its base has sides of length `7`, and its base has a corner with an angle of `( pi)/6`. What is the pyramid's surface area?

Answer 1

Total Surface Area T S A = `color (purple)(545.9996)`

Explanation:

AB = BC = CD = DA = a = 7
Height OE = h = *5
OF = a/2 = 7/2
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(5^2+((7/2)^2)) = color(red)37.25`

Area of `DCE = (1/2)*a*EF = (1/2)*7*37.25== color(red)(103.375)`
Lateral surface area `= 4*Delta DCE = 4*103.375 = color(blue)(521.5)`

`/_C = pi/6, /_C/2 = pi/12`
diagonal `AC = d_1` & diagonal `BD = d_2`
`OB = d_2/2 = BC*sin (C/2)=7*sin(pi/12) = **1.8117**`

`OC = d_1/2 = BC cos (C/2) = 7* cos (pi/12) = **6.7615**`

Area of base ABCD `= (1/2)*d_1*d_2 = (1/2)(2*1.8117)(2*6.7615) = color (blue)(24.4996)`

Total Surface Area `= Lateral surface area + Base area`
T S A `=521.5 + 24.4996 = color(purple)(545.9996)`