What is the maximum volume of a pyramid that can fit inside a cube with a side length of 12 cm?

576
288
1728
864

1 answer

The maximum volume of a pyramid that can fit inside a cube occurs when the base of the pyramid is a square with sides equal to the side length of the cube. Therefore, the base of the pyramid has an area of 12 cm x 12 cm = 144 cm².

The height of the pyramid will be half the length of a diagonal of the square base. Using the Pythagorean theorem, we can find that the diagonal of the square base is:

√(12 cm)^2 + (12 cm)^2 = √288 cm² ≈ 16.97 cm

Therefore, the height of the pyramid is:

1/2 x 16.97 cm ≈ 8.49 cm

The volume of the pyramid is:

1/3 x base area x height = 1/3 x 144 cm² x 8.49 cm ≈ 1728 cm³

Therefore, the maximum volume of a pyramid that can fit inside a cube with a side length of 12 cm is 1728 cm³.

Answer: 1728.