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 `(7 pi)/8`. What is the pyramid's surface area?

Answer 1

`color(crimson)(A_(base) = L S A + A_(base) = 18.2 + 9.57 = 27.77 " sq units"`

Explanation:

`"Total Surface Area of Rhombus " T S A = "lateral Surface Area " L S A + "Area of Base " A_(base)`

`L S A = 4 * (b/2) * l, " where b is the base length and l the slant height"`

`b = 5, h = 8, theta = (7pi)/8`

`l = sqrt (h^2 + (b/2)^2) = sqrt(7^2 + (5/2)^2) = 3.64`

`L S A = 4 * (1/2) * l * (b/2) = 2 * 3.64 * 2.5 = 18.2`

`A_(base) = b^2 sin theta = 5^2 * sin ((7pi)/8) = 9.57`

`color(crimson)(A_(base) = L S A + A_(base) = 18.2 + 9.57 = 27.77 " sq units"`