The volume of a regular quadrangular pyramid is 24, the height of the pyramid is 4.

The volume of a regular quadrangular pyramid is 24, the height of the pyramid is 4. Find the area of the pyramid section by a plane passing through its top and the diagonal of the base.

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

V = h * Sosn / 3.

At the base of a regular quadrangular pyramid lies a square, which means that the area of ​​the base is equal to the square of the side of the base:

Sop = a ^ 2.

Knowing the volume of the pyramid and its height, we find the side of the base:

a ^ 2 = 3 * V / h = 3 * 24/4 = 18;

a = √18.

By the Pythagorean theorem, we find the diagonal of the base:

d ^ 2 = a ^ 2 + a ^ 2 = 18 + 18 = 36 = 62;

d = 6.

The section of the pyramid by a plane passing through its apex and the diagonal of the base is a triangle, one of the sides of which is the diagonal of the base, and the height drawn to it coincides with the height of the pyramid. We find the area of ​​this triangle as half the product of the height of the pyramid and the diagonal of the base:

Ssection = 0.5 * d * h = 0.5 * 6 * 4 = 12.