The height of a regular 4-angled pyramid is 6 cm and the formation with a side face is an angle

The height of a regular 4-angled pyramid is 6 cm and the formation with a side face is an angle of 30 degrees. Find the volume of the pyramid.

The volume of any pyramid can be found using the following formula:

V = 1 / 3SH, where S is the area of ​​the base of the pyramid, H is its height.

The base of a regular quadrangular pyramid is a square. The side face, half of the base diagonal and the height of the pyramid form a right-angled triangle between themselves, where the side face is the hypotenuse. By the Pythagorean theorem

a² = b² + H², where a is the side face and b is half the diagonal of the base.

Opposite an angle of 30º in a right-angled triangle lies a leg equal to half of the hypotenuse. Knowing this, as well as the height of the pyramid, we write the equation in a new way and solve it:

(2b) ² = b² + 6²,

4b² – b² = 36,

3b² = 36,

b² = 12,

b = √12,

b = 2√3 cm.

We have found half of the base diagonal, and now we will find its full value:

d = 2√3 * 2 = 4√3 cm.

The area of ​​the square across the diagonal is:

S = 1 / 2d².

Let’s find it:

S = ½ * (4√3) ² = ½ * 16 * 3 = 8 * 3 = 24 cm².

Now we can find the volume:

V = 1/3 * 24 * 6 = 48 cm³.

Answer: the volume of a regular quadrangular pyramid is 48 cm³.