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 `7`, its base's sides have lengths of `8`, and its base has a corner with an angle of `( pi)/4`. What is the pyramid's surface area?

Answer 1

T S A = 174.2516

Explanation:

AB = BC = CD = DA = a = 8
Height OE = h = 7
OF = a/2 = 8/2 = 4
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(7^2+4^2) = color(red)(8.0623)`

Area of `DCE = (1/2)*a*EF = (1/2)*8*8.0623 = color(red)(32.2492)`
Lateral surface area `= 4*Delta DCE = 4*32.2492 = color(blue)(128.9968)`

`/_C = pi - (3pi)/4 = (pi)/4`
Area of base ABCD `= a* a * sin /_C = 8^2 sin (pi/4) = 45.2548`

T S A `= Lateral surface area + Base area`
T S A `=128.9968 + 45.2548 = color(purple)(174.2516)`