Calculate the Diagonal of a 7×10×4 Rectangular Prism: Space Geometry Problem

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

A face diagonal goes across one rectangular face (like from corner to corner on the front face). A space diagonal goes through the entire 3D prism from one corner to the opposite corner in 3D space.

Q: Why can't I just use the regular Pythagorean theorem?

The regular Pythagorean theorem $ a^2 + b^2 = c^2 $ only works in 2D. For 3D space diagonals, you need the 3D version: $ \sqrt{a^2 + b^2 + c^2} $.

Q: How do I know which dimensions to use?

Use all three dimensions of the rectangular prism: length, width, and height. From the diagram: length=10, width=7, height=4. It doesn't matter which order you use them in the formula.

Q: Do I need to simplify √165?

Check if 165 has perfect square factors: $ 165 = 3 \times 5 \times 11 $. Since none repeat, $ \sqrt{165} $ is already in simplest form.

Q: Can I solve this step-by-step like the explanation shows?

Yes! You can find a face diagonal first ($ \sqrt{10^2 + 4^2} = \sqrt{116} $), then use it with the third dimension: $ \sqrt{116 + 7^2} = \sqrt{165} $. Both methods give the same answer!

Space Diagonal Calculation with 3D Pythagorean Theorem

Shown below is the rectangular prism $ABCDA^1B^1C^1D^1$ .

Calculate the diagonal of the rectangular prism.

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the diagonal of the box

00:03 Let's use the Pythagorean theorem in triangle CC1D1 to find D1C

00:17 Let's substitute appropriate values according to the given data and solve for D1C

00:26 This is the length of side D1C squared

00:31 Now let's use the Pythagorean theorem in triangle BCD1 to find D1B

00:43 Let's substitute appropriate values according to the given data and solve for BD1

01:05 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Shown below is the rectangular prism $ABCDA^1B^1C^1D^1$ .

Calculate the diagonal of the rectangular prism.

Step-by-step solution

Let's look at face CC1D1D and use the Pythagorean theorem to find the diagonal of the face:

$D_1C_1^2+CC_1^2=D_1C^2$

Let's insert the known data:

$10^2+4^2=D_1C^2$

$116=D_1C^2$

Let's focus a bit on triangle BCD1 and use the Pythagorean theorem to find diagonal BD1:

$D_1C^2+CB^2=BD_1^2$

Let's insert the known data:

$116+7^2=BD_1^2$

$116+49=BD_1^2$

$165=BD_1^2$

Let's find the root:

$\sqrt{165}=BD_1$

Final Answer

$\sqrt{165}$

Key Points to Remember

Essential concepts to master this topic

Formula: Space diagonal = $\sqrt{a^2 + b^2 + c^2}$ for dimensions a, b, c
Technique: Calculate $\sqrt{7^2 + 10^2 + 4^2} = \sqrt{49 + 100 + 16} = \sqrt{165}$
Check: Verify dimensions match diagram: length=10, width=7, height=4 ✓

Common Mistakes

Avoid these frequent errors

Using 2D Pythagorean theorem instead of 3D formula
Don't calculate just $\sqrt{10^2 + 7^2} = \sqrt{149}$ = wrong diagonal! This only finds the face diagonal, not the space diagonal through the entire prism. Always use all three dimensions: $\sqrt{7^2 + 10^2 + 4^2}$ .

Practice Quiz

Test your knowledge with interactive questions

Consider a right-angled triangle, AB = 8 cm and AC = 6 cm.
Calculate the length of side BC.

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 one rectangular face (like from corner to corner on the front face). A space diagonal goes through the entire 3D prism from one corner to the opposite corner in 3D space.

Why can't I just use the regular Pythagorean theorem?

The regular Pythagorean theorem $a^2 + b^2 = c^2$ only works in 2D. For 3D space diagonals, you need the 3D version: $\sqrt{a^2 + b^2 + c^2}$ .

How do I know which dimensions to use?

Use all three dimensions of the rectangular prism: length, width, and height. From the diagram: length=10, width=7, height=4. It doesn't matter which order you use them in the formula.

Do I need to simplify √165?

Check if 165 has perfect square factors: $165 = 3 \times 5 \times 11$ . Since none repeat, $\sqrt{165}$ is already in simplest form.

Can I solve this step-by-step like the explanation shows?

Yes! You can find a face diagonal first ( $\sqrt{10^2 + 4^2} = \sqrt{116}$ ), then use it with the third dimension: $\sqrt{116 + 7^2} = \sqrt{165}$ . Both methods give the same answer!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Pythagorean Theorem questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime