Please help me find the height of this pyramid?

Question

The base of the pyramid is an equilateral triangle. Each side of the base is 6.70 cm long.
The slant height of the pyramid is 3.35 cm (half of the base)
I need to find the volume of this pyramid. Please help. Thanks so much.

Answer 1

Volume is Area of the Base * Height * 1/3

6.7cm^2 = 44.89 cm ^2
44.89 * 3.35 = 150.3815
150.3815 / 3 = 50.127167

Answer 2

First of all, the base area is 6.7²*√3/4 cm².
The height of the pyramid, the apothem of the base and the slant height form a right triangle within the interior of the pyramid, where slant height is the hypotenuse. The apothem of the equilateral triangle base is 3.35/√3, so h² = 3.35² - (3.35/√3)² and h = 3.35√2/√3.

So volume = 1/3 base area *height
= 1/3 * 6.7²*√3/4 * 3.35√2/√3
= √2/3 *3.35^3 cm³ (sorry don't have a calculator handy)

Answer 3

The area of the base is 3.35^2sqrt(3) = 11.225sqrt(3).
The altitude is Sqrt(3.35^2 -3.35sqrt(3)/3) = 3.048
So volume = .33*11.225*1.732 *3.048 = 19 .6 cm^3