Question

A square pyramid has base area 64 square centimeters. The surface area of this pyramid is 224 square centimeters. Find the height (h) to the nearest tenth of a centimeter. Pyramid height=_____cm.

Answer 1

`h \approx 12.7\,cm`

Explanation:

The side of the square base is:

`l = \sqrt{64\,cm^(2)}`

`l = 8\,cm`

The formula for the surface area of the pyramid is:

`A_(s) = (1)/(2)\cdot (3\cdot l)\cdot \sqrt{0.25\cdot l^(2)+h^(2)} + l^(2)`

The height is cleared in the previous expression:

`A_(s) - l^(2) = (3)/(2)\cdot l\cdot \sqrt{0.25\cdot l^(2)+h^(2)}`

`(2)/(3)\cdot (A_(s)-l^(2))/(l) = \sqrt{0.25\cdot l^(2) + h^(2)}`

`(4)/(9)\cdot \left((A_(s)-l^(2))/(l) \right)^(2) = (1)/(4) \cdot l^(2) + h^(2)`

`h = \sqrt{(4)/(9)\cdot \left((A_(s)-l^(2))/(l) \right)^(2) - (1)/(4)\cdot l^(2)}`

`h = \sqrt{(4)/(9)\cdot \left[(224\,cm^(2)-(8\,cm)^(2))/(8\,cm)\right]^(2)-(1)/(4)\cdot (8\,cm)^(2) }`

`h \approx 12.7\,cm`