The height of a regular quadrangular pyramid is 4 cm, and the side is 5 cm. Find the volume of the pyramid.

Consider a right-angled triangle formed by the height, side and half of the diagonal (at the base – a square, the diagonals are equal and the intersection point is halved). By the Pythagorean theorem, we find half of the diagonal. Note that we get an Egyptian triangle, which means that half of the diagonal is 3. Whole diagonal = 6 cm. Find the side of the square of the base, knowing the diagonal. A square has all sides equal, which means, according to the Pythagorean theorem: 2 * (side) ^ 2 = (diagonal) ^ 2 => side = 3 roots of 2. Base area = side in a square = 18. Pyramid volume = 1/3 * base area * height = 1/3 * 18 * 4 = 24 cm ^ 3.
Answer: The volume is 24 cm ^ 3