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 has sides of length `5`, and its base has a corner with an angle of `pi/4`. What is the pyramid's surface area?

Answer 1

`"Total Surface Area of the pyramid " = color(blue)(91.98 " sq units"`

Explanation:

Steps : 1. Find the base area of the rhombus.

1. Find the area of one slant triangle of the pyramid.

2. Multiply it by 4 to find the lateral surface area.

3. Sum the base and lateral surface areas to get the total surface area.

Given : ``a = 5, h = 7, theta = pi/4#

`"Area of rhombus base " A_r = a * a sin theta = 5^2 * sin (pi/4) ~~ 17.68`

`" Slant height " l = sqrt(h^2 + (a/2)^2) = sqrt(7^2 + 2.5^2) ~~ 7.43`

`"Area of slant " Delta " " A_t = (1/2) a * l = 0.5 * 5 * 7.43 ~~ 18.575`

`"Lateral Surface Area " A_l = 4 * A-t = 4 * 18.575 = 74.3`

`"Total Surface Area of the pyramid " = A_b + A_l`

`=> 17.68 + 74.3 = color(blue)(91.98 " sq units"`