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's sides have lengths of `5`, and its base has a corner with an angle of `(5 pi)/8`. What is the pyramid's surface area?

Answer 1

T S A = 78.999

Explanation:

AB = BC = CD = DA = a = 5
Height OE = h = 5
OF = a/2 = 5/2 = 2.5
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(5^2+2.5^2) = color(red)(5.5902)`

Area of `DCE = (1/2)*a*EF = (1/2)*5*5.5902 = color(red)(13.9755)`
Lateral surface area `= 4*Delta DCE = 4*13.9755 = color(blue)(55.902)`

`/_C =pi - (5pi)/8 = (3pi)/8`
Area of base ABCD `= a* a * sin /_C = 5^2 sin ((3pi)/8) = 23.097`

T S A `= Lateral surface area + Base area`
T S A `=55.902 + 23.097 = color(purple)(78.999)`