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.
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