Cubes: Using the formula to calculate surface area

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

• Step 1: Identify the known information.
• Step 2: Apply the surface area formula for the cube.
• Step 3: Solve for the side length of the cube.

Now, let's work through each step:

Step 1: We know the total surface area of the cube is given as 24 cm².

Step 2: The formula for the surface area of a cube is:

$S = 6a^2$

where $S$ is the surface area and $a$ is the side length of the cube.

Step 3: We set the surface area equal to 24 cm² and solve for $a$:

$6a^2 = 24$

Divide both sides by 6:

$a^2 = 4$

Take the square root of both sides to solve for $a$:

$a = \sqrt{4} = 2 \text{ cm}$

Therefore, the length of each side of the cube is $2 \, \text{cm}$.

To solve this problem, we will calculate the volume and surface area of a cube with edge length 8 cm.

• Step 1: Calculate the volume of the cube using the formula $V = a^3$.

Given that the edge length $a = 8$ cm, the volume is calculated as follows:

$V = 8^3 = 8 \times 8 \times 8 = 512 \text{ cm}^3$

• Step 2: Calculate the surface area of the cube using the formula $S = 6a^2$.

Using the same edge length $a = 8$ cm, we find the surface area:

$S = 6 \times 8^2 = 6 \times (8 \times 8) = 6 \times 64 = 384 \text{ cm}^2$

Thus, the calculated volume and surface area of the cube are, respectively, 512 cm$^3$ and 384 cm$^2$.

Therefore, the correct solution to the problem, matching the given answer choices, is choice 1: $V = 512, S = 384$.

Let's solve this problem step-by-step:

Step 1: Given the surface area $S = 486$, we know the formula for the surface area of a cube is:

• $S = 6a^2$

Step 2: We need to rearrange this formula to find $a$. The equation becomes:

• $a^2 = \frac{S}{6}$
• $a = \sqrt{\frac{S}{6}}$

Step 3: Substitute the given surface area into this equation:

$a = \sqrt{\frac{486}{6}}$

Step 4: Perform the division:

$a = \sqrt{81}$

Step 5: Calculate the square root:

$a = 9$

Now that we have found the side length, let's find the volume:

Step 6: Use the formula for the volume of a cube:

• $V = a^3$

Step 7: Substitute $a = 9$ into the volume formula:

$V = 9^3$

Step 8: Calculate the cube:

$V = 729$

Thus, the length of the side of the cube is $\mathbf{9}$ and the volume of the cube is $\mathbf{729}$.

The final answer matches the given multiple choice result:

$a=9,V=729$

To solve this problem, we will follow these steps:

• Step 1: Identify the given information and formula to use.
• Step 2: Calculate the surface area using the formula.
• Step 3: Perform the arithmetic to find the answer.

Now, let's work through each step:

Step 1: We are given that the edge length of the cube is $10$ cm. The formula for the surface area of a cube is $6 \times s^2$, where $s$ is the edge length.

Step 2: Substitute the edge length into the surface area formula:

$6 \times (10)^2$

Step 3: Calculate the surface area:

$6 \times 100 = 600$

Therefore, the total surface area of the cube is $600$ cm².

To solve this problem, we will calculate the surface area of a cube with each edge measuring 9 cm using the formula for the surface area of a cube.

Step 1: Identify the given information
The edge length $s$ of the cube is 9 cm.

Step 2: Apply the formula for the surface area
The formula for the surface area $A$ of a cube is $A = 6s^2$.

Step 3: Perform the calculation
Plug the given value into the formula:

$A = 6 \times (9 \, \text{cm})^2$

Calculate the square of the side length:
$9^2 = 81 \, \text{cm}^2$

Now multiply by 6 to find the total surface area:
$A = 6 \times 81 \, \text{cm}^2 = 486 \, \text{cm}^2$

Therefore, the surface area of the cube is $486 \, \text{cm}^2$.

To calculate the surface area of the cube, follow these steps:

• Step 1: Apply the appropriate formula. The formula for the surface area of a cube is $S = 6a^2$, where $a$ is the edge length.

• Step 2: Substitute the given values into the formula. In this problem, $a = 8$ cm.

• Step 3: Perform the calculation. Replace $a$ in the formula: $\begin{aligned} S = 6 \times (8 \, \text{cm})^2 = 6 \times 64 \, \text{cm}^2 = 384 \, \text{cm}^2 \end{aligned}$

Thus, the surface area of the cube is $\mathbf{384 \, \text{cm}^2}$.

Finally, comparing the result to the given choices, choice 1 is correct: $384$ cm².

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

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula
• Step 3: Perform the necessary calculations

Now, let's work through each step:

Step 1: The problem gives us that the edge length of the cube is $7$ cm.

Step 2: We'll use the formula for the surface area of a cube, which is $6a^2$, where $a$ is the edge length.

Step 3: Substitute the given edge length into the formula:

$6 \times (7)^2 = 6 \times 49$.

Calculate $6 \times 49$:

$6 \times 49 = 294$.

Thus, the surface area of the cube is $\ 294$ cm².

However, let's ensure accuracy with the problem's indicated final answer:

The problem indicates the correct final answer's intention should be $\ 294$ cm².

Therefore, the surface area of the cube is indeed $294$ cm²..

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

• Step 1: Identify the given edge length of the cube.
• Step 2: Apply the surface area formula for a cube.
• Step 3: Calculate the surface area using the given edge length.

Now, let's work through each step:
Step 1: The edge length of the cube is given as $6$ cm.
Step 2: We'll use the formula for the surface area of a cube: $S = 6a^2$, where $a$ is the edge length.
Step 3: Plugging in the value, we calculate: $S = 6 \times 6^2 = 6 \times 36 = 216$ cm².

Therefore, the surface area of the cube is $216$ cm².

To determine the surface area of the cube, we will follow a systematic approach:

• Step 1: Identify the provided information.
  The edge length of the cube is given as $5$ cm.

• Step 2: Utilize the formula for the surface area of a cube.
  The formula is $6a^2$, with $a$ representing the edge length of the cube.

• Step 3: Substitute the given edge length into the formula and perform the calculation.
  Using $a = 5$ cm, the calculation becomes:
  $6 \times (5 \, \text{cm})^2 = 6 \times 25 \, \text{cm}^2 = 150 \, \text{cm}^2.$

The computed surface area of the cube is therefore $150$ cm².

Therefore, the correct choice among the given options is the fourth choice, which is $150 ~cm²$.

In order to find the surface area of the cube, we need to understand that a cube has 6 identical square faces.

The formula for the surface area of a cube is given by:

$A = 6s^2$

where $s$ is the length of an edge of the cube.

According to the problem, each edge of the cube is 4 cm long. We can substitute this value into the formula:

$A = 6 \times (4)^2$

This simplifies to:

$A = 6 \times 16$

Performing the multiplication gives us:

$A = 96$

Thus, the total surface area of the cube is $96 \, \text{cm}^2$.

Therefore, the correct answer is $\text{96 cm}^2$, which corresponds to choice 2.

To calculate the surface area of a cube, we use the formula:

$\text{Surface Area} = 6a^2$

where $a$ is the length of each edge of the cube.

Given $a = 3 \text{ cm}$, we substitute this value into the formula:

$\text{Surface Area} = 6 \times (3)^2$

$= 6 \times 9$

$= 54 \text{ cm}^2$

Therefore, the surface area of the cube is $54 \text{ cm}^2$.

This matches choice 4.

To solve this problem, we will calculate the surface area of the cube with edge length 11 cm using the formula for the surface area of a cube. The steps are as follows:

• Step 1: Identify the side length of the cube
  The problem states that each edge of the cube is 11 cm long.
• Step 2: Apply the formula for the surface area of a cube
  The surface area $A$ of a cube is given by the formula $A = 6s^2$, where $s$ is the side length.
• Step 3: Perform the calculation
  Substitute $s = 11$ cm into the formula:

$A = 6 \times 11^2$

Calculate $11^2 = 121$:

$A = 6 \times 121$

Calculate $6 \times 121 = 726$.

Therefore, the surface area of the cube is $\textbf{726 cm}^2$.

Looking at the choices provided, the correct answer is the third choice: $726 \text{ cm}^2$.

Therefore, the solution to the problem is $\textbf{726 cm}^2$.

To solve for the surface area of the cube, follow these steps:

• Step 1: Identify the given information. The edge length of the cube is 12 cm.
• Step 2: Apply the surface area formula for a cube, which is $A = 6s^2$, where $s$ is the side length.
• Step 3: Substitute the side length into the formula: $A = 6 \times 12^2$.
• Step 4: Calculate $12^2 = 144$.
• Step 5: Multiply the result by 6: $A = 6 \times 144 = 864$.

Therefore, the surface area of the cube is $864 \, \text{cm}^2$.

To solve this problem, we'll determine the surface area using the formula for a cube:

• Step 1: Identify the given information — Each side of the cube is $X$.
• Step 2: Recall the surface area formula — For a cube, it is $6 \times (\text{side length})^2$.
• Step 3: Substitute the side length $X$ into the formula.

Now, let's proceed with the solution:
Step 1: The cube has each side $= X$.
Step 2: The surface area formula for a cube is given by $6 \times (\text{side length})^2$.
Step 3: Substituting $X$ for the side length, we find the surface area is $6 \times X^2$.

Therefore, the surface area of the cube in terms of $X$ is $\mathbf{6X^2}$.