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

Answer 1

Total Surface Area

`color(green)( T S A = A_T = A_L + A_R = 23.097 + 74.3303 = 97.4273)`

Explanation:

Area of Rhombus base `A_R = a * a sin theta`

`A_R = 5 * 5 * sin ((5pi)/8) ~~ 23.097`

Area of slant triangle `A_t = (1/2) a * h_l` where `color(red)(a)` is the rhombus base length and `color(red)(h_l)`is the slant height of the triangle.

`h_l = sqrt(h^2 + (a/2)^2)` where `color(red)(h)` is the pyramid’s height.

Lateral Surface Area `A_L = 4 * A_t = 4 * (1/2) * 5 * sqrt(7^2 + (5/2)^2)`

`L S A = A_L = 74.3303`

Total Surface Area `color(green)( T S A = A_T = A_L + A_R = 23.097 + 74.3303 = 97.4273)`