A regular triangular pyramid has slant height 6 cm and base edges of length 15 cm. Find:
Volume of the pyramid?

V ≈ 134.883 cm³

Step-by-step explanation:

Volume of a pyramid is:

V = ⅓ A h

where A is the area of the base and h is the height.

The base is an equilateral triangle.  Its area can be found with:

A = ½ aP

where a is the apothem and P is the perimeter.

The height of the pyramid can be found with Pythagorean theorem:

L² = h² + a²

h = √(L² − a²)

So the volume is:

V = ⅙ aP √(L² − a²)

V = ⅙ a(3s) √(L² − a²)

V = ½ as √(L² − a²)

We know L = 6 cm and s = 15 cm.  All we have to do now is find the apothem.  We can do that using properties of 30-60-90 triangles.

√3 a = s/2

a = s/(2√3)

a² = s²/12

The volume is:

V = ½ (s/(2√3)) s √(L² − s²/12)

V = s²/(4√3) √(L² − s²/12)

V = ¼ s² √(L²/3 − s²/36)

Plugging in values:

V = ¼ (15 cm)² √((6 cm)²/3 − (15 cm)²/36)

V = 56.25 cm² √(12 cm² − 6.25 cm²)

V = 56.25 cm² √(5.75 cm²)

V ≈ 134.883 cm³

$12. that is my answer.