Calculate the Space Diagonal: Finding the Corner-to-Corner Distance in a 10cm Cube

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

A face diagonal goes across a square face of the cube, while a space diagonal goes from one corner to the opposite corner through the inside of the cube. The space diagonal is always longer!

Q: Why do we use √3 instead of √2 for cubes?

Because we're moving in three dimensions (length, width, height). The formula $ d = a\sqrt{3} $ comes from applying the Pythagorean theorem twice: first for the face diagonal, then adding the third dimension.

Q: How do I remember the space diagonal formula?

Think "3D = √3"! For any cube, the space diagonal is always the edge length times $ \sqrt{3} \approx 1.73 $. It's about 1.7 times longer than each edge.

Q: Why is 17.3 cm the right answer and not exactly 17.32?

The problem asks for a practical measurement, so we round to one decimal place. $ 10\sqrt{3} = 17.320... $ rounds to 17.3 cm for real-world precision.

Space Diagonal with Three-Dimensional Pythagorean Theorem

Shown below is a cube with edges equal to 10 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

00:14 We'll substitute appropriate values and solve to find the diagonal

00:35 This is the diagonal in triangle DD'C

00:42 We'll use the Pythagorean theorem in triangle D'A'C

00:48 We'll substitute appropriate values and solve to find the diagonal

01:14 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 equal to 10 cm.

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

Step-by-step solution

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.

Final Answer

$17.3$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Space diagonal equals edge length times square root of three
Technique: Calculate $d = 10\sqrt{3} = 10 \times 1.732 = 17.32$ cm
Check: Diagonal should be longer than face diagonal $10\sqrt{2} \approx 14.1$ cm ✓

Common Mistakes

Avoid these frequent errors

Using 2D Pythagorean theorem instead of 3D formula
Don't calculate $\sqrt{10^2 + 10^2} = 10\sqrt{2} \approx 14.1$ cm = face diagonal only! This ignores the third dimension and gives a shorter distance. Always use $d = a\sqrt{3}$ for space diagonals connecting opposite corners.

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 space diagonal?

A face diagonal goes across a square face of the cube, while a space diagonal goes from one corner to the opposite corner through the inside of the cube. The space diagonal is always longer!

Why do we use √3 instead of √2 for cubes?

Because we're moving in three dimensions (length, width, height). The formula $d = a\sqrt{3}$ comes from applying the Pythagorean theorem twice: first for the face diagonal, then adding the third dimension.

How do I remember the space diagonal formula?

Think "3D = √3"! For any cube, the space diagonal is always the edge length times $\sqrt{3} \approx 1.73$ . It's about 1.7 times longer than each edge.

What if the cube has different edge lengths?

Then it's not a cube - it's a rectangular prism! For cubes, all edges must be equal. If they're different, you need the more complex formula $d = \sqrt{a^2 + b^2 + c^2}$ .

Why is 17.3 cm the right answer and not exactly 17.32?

The problem asks for a practical measurement, so we round to one decimal place. $10\sqrt{3} = 17.320...$ rounds to 17.3 cm for real-world precision.

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