Cube Volume and Surface Area Practice Problems with Solutions

To solve this problem, we'll follow these steps:

• Step 1: Identify the base area of the cube.
• Step 2: Use the formula to find the side length of the base.
• Step 3: Recognize that the height of the cube is equal to the side length of the base.

Now, let's work through each step:
Step 1: We are given the base area of the cube as $16 \, \text{cm}^2$.
Step 2: The area of a square is calculated using the formula $\text{side}^2$, where "side" is the length of each side of the square.
Setting up the equation: $\text{side}^2 = 16$. Solving for the "side," we find $\text{side} = \sqrt{16} = 4 \, \text{cm}$.
Step 3: Since the cube is a regular geometric shape, the height is equal to the side length of the base. Therefore, the height of the cube is $4 \, \text{cm}$.

Therefore, the height of the cube is $4 \, \text{cm}$.

To solve this problem, we'll follow these steps:

• Step 1: Understand the relationship between the base area and the side length of a cube.
• Step 2: Calculate the side length using the square area formula.
• Step 3: Conclude that the height of the cube is equal to this side length.

Now, let's work through each step:

Step 1: The basic property of a cube is that all of its three dimensions (length, width, and height) are equal. We know the base area of this cube is given as 36 cm².

Step 2: Using the formula for the area of a square, we have $s^2 = 36$, where $s$ is the side length of the base.

Solving for $s$, we find:

$s = \sqrt{36} = 6 \, \text{cm}$

Step 3: Since all sides of a cube are equal, the height of the cube is also $6 \, \text{cm}$.

Therefore, the height of the cube is $6 \, \text{cm}$.

To solve this problem, we'll follow these steps:

• Identify the given information: The edge of the cube is 3 cm.
• Apply the formula for the volume of a cube: $V = a^3$.
• Calculate the volume by substituting the given edge length into the formula.

Now, let's work through each step:

Step 1: The edge length $a$ is 3 cm.

Step 2: The formula for the volume of a cube is $V = a^3$. Substituting the given edge length, we have:

$V = 3^3$

Step 3: Calculate $3^3$:

$3 \times 3 \times 3 = 27$

Therefore, the volume of the cube is $27$ cubic centimeters.

Thus, the solution to the problem is $27$ cm$^3$.

To determine if we can calculate the volume of the cube, let's start by analyzing the given information:

1. The base area of the cube is given as $9 \, \text{cm}^2$. In a cube, each face is a square, so this area corresponds to the area of one face.
2. To find the side length $s$ of the square face, use the formula for the area of a square: $A = s^2$.
3. Set up the equation based on the given area: $s^2 = 9$.
4. Solve for $s$ by taking the square root of both sides: $s = \sqrt{9} = 3 \, \text{cm}$.
5. Now that we have the side length $s$, calculate the volume $V$ of the cube using the formula for the volume of a cube: $V = s^3$.
6. Substitute $s = 3 \, \text{cm}$ into the volume formula: $V = 3^3 = 27 \, \text{cm}^3$.

Therefore, the volume of the cube is $27 \, \text{cm}^3$.

Among the given choices, the correct answer is:

• Choice 3: $27$

To find the sum of the lengths of all the edges of a cube, we can follow these steps:

• Step 1: Recognize that a cube has 12 edges, and each edge is the same length.
• Step 2: Given the side length of the cube is 4 cm, use the formula for the total edge length.

The formula for the total length of the edges of a cube is:

$\text{Total length} = \text{number of edges} \times \text{length of one edge}$

Substituting the known values, we have:

$\text{Total length} = 12 \times 4 \, \text{cm}$

Calculating this gives:

$\text{Total length} = 48 \, \text{cm}$

Therefore, the sum of the lengths of the cube's edges is $48 \, \text{cm}$.