Cubes: Finding a diagonal - using the Pythagorean theorem

To solve this problem, we need to determine the length of the diagonal of one face of a cube with edge length 5 cm.

• Step 1: Recognize that one face of the cube is a square with side length 5 cm.
• Step 2: Apply the Pythagorean theorem formula for the diagonal $d$ of a square, $d = s\sqrt{2}$, where $s$ is the side length.
• Step 3: Substitute $s = 5$ cm into the formula to compute the diagonal.

Now, let's perform the calculations:
The diagonal $d$ of the square (face of the cube) is given by:

$d = 5\sqrt{2}$.

Therefore, the length of the diagonal on the cube's face is $5\sqrt{2}$ cm.

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

• Step 1: Identify the side length of the square face on the cube.
• Step 2: Use the appropriate formula to find the diagonal of a square.
• Step 3: Calculate using the given measurements.

Now, let's work through these steps:
Step 1: The side length of the square face on the cube is given as 2 cm.
Step 2: The formula for the diagonal $d$ of a square with side length $s$ is $d = s\sqrt{2}$.
Step 3: Substituting the given side length into the formula, we get:

$d = 2 \times \sqrt{2}$

Therefore, the length of the diagonal of the cube's face is $2\sqrt{2}$ cm.

The correct multiple-choice answer that corresponds to our solution is:

• Choice 4: $2\times\sqrt{2}$ cm

Therefore, the solution to the problem is $2\sqrt{2}$ cm.

To solve for the diagonal of a face of the cube, follow these steps:

• Identify the side length of the square face: Each side of the face of the cube is 4 cm.
• Recognize that the diagonal forms a right triangle with two sides of the square face.
• Apply the Pythagorean theorem: The diagonal $d$ is found using $d = \sqrt{a^2 + b^2}$ where $a$ and $b$ are the sides of the square.
• Substitute the side lengths into the formula: Since both $a$ and $b$ are 4 cm, the formula becomes $d = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32}$.
• Simplify the square root: $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4 \times \sqrt{2}$.

Therefore, the length of the diagonal of the cube's face is $4 \times \sqrt{2}$ cm.

To find the inner diagonal of a cube where each edge is 10 cm long, we will use the Pythagorean theorem in three dimensions. For a cube with edge length $a$, the formula to find the diagonal $d$ is:

$d = a \sqrt{3}$

In this problem, the edge length $a = 10$ cm. Substituting into the formula gives:

$d = 10 \sqrt{3}$

Calculating $10 \sqrt{3}$, we estimate $\sqrt{3} \approx 1.732$. Thus:

$d \approx 10 \times 1.732 = 17.32$

Rounded to one decimal place (as often required for physical measures), the length of the inner diagonal of the cube is approximately $17.3$ cm.

Therefore, the correct choice is the one that matches this calculation, which is:

$17.3$ cm.

To find the length of the inner diagonal of a cube, we'll use the formula for the main space diagonal of a cube, which can be derived using the Pythagorean theorem:

The formula for the space diagonal ($d$) of a cube with edge length $a$ is:

$d = \sqrt{a^2 + a^2 + a^2}$.

Given that each side of the cube is 6 cm, substitute $a = 6$ cm into the formula:

$d = \sqrt{6^2 + 6^2 + 6^2} = \sqrt{3 \times 6^2} = \sqrt{3 \times 36} = \sqrt{108}$.

Now, calculate the square root of 108:

$\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}$.

Using a calculator or an estimated value for $\sqrt{3} \approx 1.732$, we calculate:

$6\sqrt{3} \approx 6 \times 1.732 = 10.392$.

Therefore, the length of the inner diagonal of the cube is approximately $10.39$ cm.

The correct choice for this problem is option 1: $10.39$ cm.