Answer:
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
