The volume of a cube is calculated by multiplying the side length by itself three times (s * s * s), which results in s^3. This formula is fundamental for computing the space occupied by a cube.

The surface area of a sphere is given by the formula 4πr^2, which accounts for the entire area covering the sphere. This is a key formula in geometry.

The volume of a rectangular prism is obtained by multiplying its length, width, and height (l * w * h). This represents the space contained within it.

The volume of a cylinder is calculated using the formula πr^2h, which multiplies the area of the circular base (πr^2) by the height (h). This formula helps determine the amount of space inside the cylinder.

The surface area of a cube is calculated by finding the area of one face (s^2) and multiplying by six, since a cube has six identical faces. Hence, the formula is 6s^2.

By applying the surface area formula for a cube, 6s², we compute 6*(4²) which equals 96 cm². This problem reinforces the understanding of area calculation for cubes.

Using the volume formula for a cylinder, πr²h, and substituting 3 cm for r and 10 cm for h, we get 90π cm³. This problem emphasizes substituting and computing correctly.

By inserting r = 2 cm into the volume formula for a sphere, (4/3)πr³, we calculate the volume as (32/3)π cm³. This reinforces the use of the sphere volume formula.

We apply the surface area formula for a cylinder, adding the areas of the two circular bases and the lateral area: 2πr² + 2πrh. This yields 120π cm² for the given dimensions.

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Here, 3*4*5 gives a volume of 60 cm³.

When all dimensions of a solid are scaled by a factor of k, the new volume becomes k³ times greater than the original volume. This principle is derived from the cubic relationship in volume calculations.

The volume of a cone is given by (1/3)πr²h. Replacing r with 6 cm and h with 9 cm, we obtain 108π cm³. This problem emphasizes careful substitution into the formula.

The lateral surface area of a cylinder refers to the curved surface area and is calculated as 2πrh, where r is the radius and h is the height. This formula excludes the areas of the circular bases.

First, convert the diameter to radius (10/2 = 5 cm). Then substitute into the volume formula for a sphere, (4/3)πr³, to find (500/3)π cm³. This reinforces conversion between diameter and radius.

The surface area of a cube is 6a². By setting 6a² equal to 150, solving for a² gives 25, hence the edge length a is 5 units. This illustrates basic algebraic manipulation with geometric formulas.

The cube's volume is 64 cm³ and the attached hemisphere's volume is (16/3)π cm³. Adding these gives the total volume, ensuring that different shapes' formulas are applied correctly in composite solids.

The solution involves adding the lateral surface area of the cylinder (60π), its bottom base area (9π), and the lateral surface area of the cone (3π√34). The shared base between the two shapes is excluded, leading to a total surface area of 69π + 3π√34.

By rearranging the pyramid volume formula, Volume = (1/3) × base area × height, we find that the base area is (96 × 3) / 8, which equals 36 cm². This demonstrates the application of volume formulas to find missing dimensions.

With a diameter of 8 m, the radius is 4 m. Substituting into the cylinder volume formula, Volume = πr²h, we get 3.14 × 16 × 12 ≈ 602.88 m³, which rounds to approximately 603 m³. This question combines geometry with arithmetic estimation.

Since the volume of a cone is one-third that of a cylinder with the same base and height, the cylinder's volume is three times larger. This reinforces the critical difference between these volume formulas.