The height of a regular quadrangular pyramid is 12 dm, the diagonal of its base is 10 √ 2, calculate the volume of the pyramid.

1. It is known that the volume of a pyramid is equal to one third of the product of the base area by its height.

2. At the base of a regular quadrangular pyramid lies a square, so we find the area S of a square by the formula

S = a², where a is the side of the square.

Let us calculate a² if, by the problem statement, the diagonal is d = 10 √2.

We know that the diagonals of the square are perpendicular and are divided in half at the intersection point, so we define a² by the Pythagorean theorem

a² = (d / 2) ² + (d / 2) ² = (5 √2) ² + (5 √2) ² = 25 * 2 + 25 * 2 = 100.

3. With the known value of the height h = 12 dm and the found area of ​​the base, we find the volume V of the pyramid

V = 1/3 * a² * h = 1/3 * 100 * 12 = 400 dm³.

Answer: The volume is 400 dm³.