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.

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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 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

Look at the triangle in the diagram. How long is side AB?

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!

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