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 `2`, its base has sides of length `7`, and its base has a corner with an angle of `(2 pi)/3`. What is the pyramid's surface area?

Answer 1

T S A = 98.8696

Explanation:

AB = BC = CD = DA = a = 7
Height OE = h = 2
OF = a/2 = 7/2 = 3.5
`EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(2^2+3.5^2) = color(red)(4.0311)`

Area of `DCE = (1/2)*a*EF = (1/2)*7*4.0311 = color(red)(14.1089)`
Lateral surface area `= 4*Delta DCE = 4*14.1089 = color(blue)(56.4356)`

`/_C = pi - (2pi)/3 = (pi)/3`
Area of base ABCD `= a* a * sin /_C = 7^2 sin (pi/3) = 42.434`

T S A `= Lateral surface area + Base area`
T S A `=56.4356 + 42.434 = color(purple)(98.8696)`