Calculate the Space Diagonal: Finding the Longest Distance Inside a 6-cm Cube

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

A face diagonal goes across a square face using $ a\sqrt{2} $. The space diagonal goes through the cube's center from corner to opposite corner using $ a\sqrt{3} $.

Q: Can I just memorize that the space diagonal equals edge × √3?

Yes! For any cube, the space diagonal is always $ a\sqrt{3} $. But understanding why helps you solve similar problems with rectangular boxes using $ \sqrt{l^2 + w^2 + h^2} $.

Q: How do I calculate √3 without a calculator?

You can estimate: $ \sqrt{3} \approx 1.7 $ (since $ 1.7^2 = 2.89 $). For better accuracy, use $ \sqrt{3} \approx 1.732 $.

Q: Will this formula work for any cube size?

Absolutely! Whether your cube has 2 cm edges or 20 cm edges, just multiply the edge length by $ \sqrt{3} \approx 1.732 $ to get the space diagonal.

Space Diagonal with 3D Pythagorean Theorem

Shown below is a cube with edges that equal 6 cm.

What is the length of the inner diagonal of the cube?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the internal diagonal in the cube

00:04 We'll use the Pythagorean theorem in triangle DD'C

00:15 The side length according to the given data

00:19 We'll substitute appropriate values and solve to find the diagonal

00:33 This is the diagonal in triangle DD'C

00:37 We'll use the Pythagorean theorem in triangle D'A'C

00:45 We'll substitute appropriate values and solve to find the diagonal

01:10 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Shown below is a cube with edges that equal 6 cm.

What is the length of the inner diagonal of the cube?

Step-by-step solution

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.

Final Answer

$10.39$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Space diagonal = $\sqrt{a^2 + a^2 + a^2} = a\sqrt{3}$
Technique: For 6 cm cube: $\sqrt{6^2 + 6^2 + 6^2} = 6\sqrt{3}$
Check: $6\sqrt{3} \approx 6 \times 1.732 = 10.39$ cm ✓

Common Mistakes

Avoid these frequent errors

Using only two dimensions instead of three
Don't use $\sqrt{6^2 + 6^2} = 6\sqrt{2} \approx 8.49$ cm! This gives the face diagonal, not the space diagonal. Always use all three dimensions: $\sqrt{a^2 + a^2 + a^2}$ for the inner diagonal through the cube's center.

Practice Quiz

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Identify the correct 2D pattern of the given cuboid:

FAQ

Everything you need to know about this question

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

A face diagonal goes across a square face using $a\sqrt{2}$ . The space diagonal goes through the cube's center from corner to opposite corner using $a\sqrt{3}$ .

Why do we use three identical terms in the formula?

The space diagonal forms the hypotenuse of a 3D right triangle. We need all three dimensions (length, width, height) which are equal in a cube, so we get $\sqrt{a^2 + a^2 + a^2}$ .

Can I just memorize that the space diagonal equals edge × √3?

Yes! For any cube, the space diagonal is always $a\sqrt{3}$ . But understanding why helps you solve similar problems with rectangular boxes using $\sqrt{l^2 + w^2 + h^2}$ .

How do I calculate √3 without a calculator?

You can estimate: $\sqrt{3} \approx 1.7$ (since $1.7^2 = 2.89$ ). For better accuracy, use $\sqrt{3} \approx 1.732$ .

Will this formula work for any cube size?

Absolutely! Whether your cube has 2 cm edges or 20 cm edges, just multiply the edge length by $\sqrt{3} \approx 1.732$ to get the space diagonal.

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