A square pyramid has a base of total area of 144 m square root 2 and a volume of 384m cubic what is the slant height of the pyramid?

Slant height of the pyramid = 11.66 m

Step-by-step explanation:

A squad pyramid has a volume of

V = ⅓ × (area of base) × (vertical height)

Area of the pyramid's square base = 144 m²

Volume = 384 m³

384 = ⅓ × 144 × (vertical height)

Vertical height = (3×384)/(144) = 8 m

But, the slant height of the square based pyramid forms a right angled triangle with the vertical height of the pyramid and the distance from the centre of the base of the pyramid to an edge of the pyramid.

The distance from the centre of the base of the pyramid to an edge of the pyramid = half of the length of the diagonal of the square base of the pyramid.

The diagonal of the square base of the pyramid also forms a right angled triangle with two sides of the square base of the pyramid.

Area of the square base of the pyramid = 144 m²

Area of a square = (side length)²

Side length = √(Area of the square) = √144 = 12 m

Using Pythagoras theorem,

(Length of the diagonal)² = 12² + 12² = 288

Length of the diagonal = (12√2) m

Half of the length of the diagonal of the square base = 6√2 m

Using Pythagoras theorem further

(Slant height of the pyramid)² = (vertical height of the pyramid)² + (Half of the length of the diagonal of the square base)²

(Slant height of the pyramid)² = 8² + (6√2)² = 64 + 72 = 136

Slant height of the pyramid = √136 = 11.66 m

Hope this Helps!!!

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