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