Calculate the Face Diagonal of a 2cm Cube: Geometric Measurement Problem

Q: What's the difference between a face diagonal and a space diagonal?

A face diagonal goes across one square face of the cube (like corner to corner on one side). A space diagonal goes through the entire cube from one corner to the opposite corner. Face diagonals use $ d = s\sqrt{2} $.

Q: Why do we multiply by √2?

The face of a cube is a square, and the diagonal of any square creates two right triangles. Using the Pythagorean theorem: $ d^2 = s^2 + s^2 = 2s^2 $, so $ d = s\sqrt{2} $.

Q: Can I just measure the diagonal instead of calculating?

While measuring might seem easier, calculating gives you the exact answer! Plus, in math problems, you need to show your understanding of geometric relationships and formulas.

Q: How do I simplify √2 in my final answer?

Don't try to convert $ \sqrt{2} $ to a decimal unless asked! Leave it as $ 2\sqrt{2} $ cm because this is the exact answer. Decimals like 2.83 are just approximations.

Face Diagonals with Square Geometry

Shown below is a cube with edges equal to 2 cm.

What is the length of the diagonal of the cube's face shown in the figure?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the diagonal of the cube face

00:04 Let's use the Pythagorean theorem in triangle ABD

00:11 We'll substitute appropriate values and solve for the diagonal

00:35 Let's break down the square root of 8 into square root of 4 times square root of 2

00:38 Let's calculate the square root

00:44 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Shown below is a cube with edges equal to 2 cm.

What is the length of the diagonal of the cube's face shown in the figure?

Step-by-step solution

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.

Final Answer

$2\times\sqrt{2}$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Diagonal of a square equals side length times √2
Technique: For 2 cm cube face: d = 2 × √2 cm
Check: Pythagorean theorem: 2² + 2² = (2√2)² = 8 ✓

Common Mistakes

Avoid these frequent errors

Using space diagonal formula instead of face diagonal
Don't use d = s√3 for face diagonals = wrong dimension! That formula is for the cube's space diagonal (corner to opposite corner through the center). Always use d = s√2 for face diagonals on square faces.

Practice Quiz

Test your knowledge with interactive questions

A cube has a total of 14 edges.

FAQ

Everything you need to know about this question

What's the difference between a face diagonal and a space diagonal?

A face diagonal goes across one square face of the cube (like corner to corner on one side). A space diagonal goes through the entire cube from one corner to the opposite corner. Face diagonals use $d = s\sqrt{2}$ .

Why do we multiply by √2?

The face of a cube is a square, and the diagonal of any square creates two right triangles. Using the Pythagorean theorem: $d^2 = s^2 + s^2 = 2s^2$ , so $d = s\sqrt{2}$ .

Can I just measure the diagonal instead of calculating?

While measuring might seem easier, calculating gives you the exact answer! Plus, in math problems, you need to show your understanding of geometric relationships and formulas.

What if the cube has different edge lengths?

The same formula works! Just substitute the new side length. If edges are 5 cm, then face diagonal = $5\sqrt{2}$ cm. The relationship stays the same for any square.

How do I simplify √2 in my final answer?

Don't try to convert $\sqrt{2}$ to a decimal unless asked! Leave it as $2\sqrt{2}$ cm because this is the exact answer. Decimals like 2.83 are just approximations.

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