determine the volume of a right rectangular pyramid with the base dimensions 6 ft by 12 ft. For each triangular face, the equal sides have length 18 ft.​ Write answer to nearest cubic ft.​

9514 1404 393

  401 ft³

Explanation:

Consider the cross section perpendicular to the base through opposite corners of the base. It will be an isosceles triangle with legs of 18 ft and a base dimension equal to the diagonal of the base rectangle.

That diagonal length is given by the Pythagorean theorem as ...

  d² = 6² +12² = 180

  d = √180 = 6√5

Then the right triangle that tells us the height of the pyramid has a base length of (1/2)(6√5) and a hypotenuse of 18. The pyramid height is ...

  (3√5)² + h² = 18²

  h = √(324 -45) = √279 = 3√31

So, the volume of the pyramid is ...

  V = (1/3)Bh

  V = (1/3)((6 ft)(12 ft))(3√31 ft) = 72√31 ft³ ≈ 401 ft³