Find the volume of a regular quadrangular pyramid, the diagonal of the base of which is 4 cm

Find the volume of a regular quadrangular pyramid, the diagonal of the base of which is 4 cm, and the side edge forms an angle of 45 degrees with the base plane.

The volume of a regular quadrangular pyramid is defined as a third of the product of the base area by the height:

V = Sosn * h / 3.

The base of a regular quadrangular pyramid is a square, the diagonal of which is 4 cm.The diagonal of the square and the two sides form a right-angled triangle, therefore:

a ^ 2 + a ^ 2 = d ^ 2;

a ^ 2 = d ^ 2/2 = 16/2 = 8.

The area of ​​the square is equal to the square of the side, which means:

Sbn = a ^ 2 = 8 cm2.

The diagonals of the square at the intersection point are halved, and the top of the pyramid is projected to this point. The side edge of the pyramid, the height and half of the diagonal form a right-angled triangle. The angle between the lateral edge and the plane of the base is 45 °, therefore, this triangle is also isosceles, which means that half of the diagonal of the base is equal to the height of the pyramid:

h = d / 2 = 4/2 = 2 cm.

V = Sbn * h / 3 = 8 * 2/3 = 16/3 = 5.33 cm3.