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

Answer 1

T S A = 167.5196

Explanation:

AB = BC = CD = DA = a = 7
Height OE = h = 8
OF = a/2 = 7/2 = 3.5
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(8^2+3.5^2) = color(red)(8.7321)`

Area of `DCE = (1/2)*a*EF = (1/2)*7*8.7321 = color(red)(30.5624)`
Lateral surface area `= 4*Delta DCE = 4*30.5624 = color(blue)(122.2496)`

`/_C = (3pi)/8`
Area of base ABCD `= a* a * sin /_C = 7^2 sin ((3pi)/8) = 45.2701`

T S A `= Lateral surface area + Base area`
T S A `=122.2496 + 45.2701 = color(purple)(167.5196)`