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

Answer 1

T S A = 108.9464

Explanation:

AB = BC = CD = DA = a = 7
Height OE = h = 5
OF = a/2 = 7/2 = 3.5
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(5^2+(7/2)^2) = color(red)(6.1033)`

Area of `DCE = (1/2)*a*EF = (1/2)*7*6.1033 = color(red)(21.3616)`
Lateral surface area `= 4*Delta DCE = 4*21.3616 = color(blue)(84.4464)`

`/_C = (pi) - ((5pi)/6) = (pi)/6`
Area of base ABCD `= a* a * sin /_C = 7^2 sin (pi/6) = 24.5`

T S A `= Lateral surface area + Base area`
T S A `=84.4464 + 24.5 = color(purple)(108.9464)`