Find the Face Diagonal of a Rectangular Prism with Base Diagonal X and Length 3X

Q: Why can't I just use X as the side of the square base?

Because X is the diagonal of the square base, not its side! The diagonal is always $ \sqrt{2} $ times longer than the side. Using X directly would give you a completely wrong prism.

Q: Which face diagonal is the question asking for?

It's asking for the diagonal of a rectangular face - the face that has one side equal to the square's side length and the other side equal to the prism's height (3X).

Q: How do I remember the relationship between square side and diagonal?

Think of a square split by its diagonal into two right triangles. Using Pythagorean theorem: $ s^2 + s^2 = d^2 $, so $ d = s\sqrt{2} $.

Q: Why is the answer $ X\sqrt{9.5} $ instead of a simpler form?

Because when you calculate $ \sqrt{\frac{X^2}{2} + 9X^2} = X\sqrt{\frac{1}{2} + 9} = X\sqrt{9.5} $, the 9.5 under the square root cannot be simplified further into a perfect square.

Q: Can I solve this without using the Pythagorean theorem?

No! Since we need the diagonal of a rectangle, we must use the Pythagorean theorem. The diagonal is the hypotenuse of a right triangle formed by the rectangle's sides.

3D Geometry with Square Base Diagonals

A rectangular prism has a square base with a diagonal length of X.

The prism has a length of 3X.

How long is the diagonal of the rectangular face of the prism.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the diagonal length of the rectangular faces in the box

00:03 Box is square according to the given data, therefore sides are equal

00:07 In a square all angles are right angles

00:12 We'll use the Pythagorean theorem in triangle A1D1C1 to find A1C1

00:15 We'll substitute appropriate values according to the given data and solve to find X

00:22 This is the length of diagonal face X

00:28 We'll express Y using X

00:38 Each face in the box is a rectangle, therefore all angles are right angles

00:47 We'll use the Pythagorean theorem in triangle AA1D1 to find AD1

00:58 We'll substitute appropriate values according to the given data and solve to find AD1

01:18 Group terms

01:22 Take the square root

01:26 This is the length of the diagonal of the rectangular faces in the box

01:29 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

A rectangular prism has a square base with a diagonal length of X.

The prism has a length of 3X.

How long is the diagonal of the rectangular face of the prism.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Determine the side length of the square base
Step 2: Calculate the diagonal of the rectangular face using the Pythagorean theorem

Now, let's work through each step:
Step 1: The side length $s$ of the square base is calculated from its diagonal $X$ as $s = \frac{X}{\sqrt{2}}$ .
Step 2: The diagonal $d$ of the rectangular face is found using $d^2 = s^2 + (3X)^2$ , which simplifies to $d = X\sqrt{9.5}$ .

The solution to the problem is $x\sqrt{9.5}$ .

Final Answer

$x\sqrt{9.5}$

Key Points to Remember

Essential concepts to master this topic

Base Analysis: Square diagonal X means side length is $\frac{X}{\sqrt{2}}$
Pythagorean Method: Face diagonal = $\sqrt{s^2 + (3X)^2} = \sqrt{\frac{X^2}{2} + 9X^2}$
Verification: Check that $X\sqrt{9.5} = X\sqrt{\frac{19}{2}}$ matches your calculation ✓

Common Mistakes

Avoid these frequent errors

Using X directly as the side length instead of converting from diagonal
Don't treat the base diagonal X as a side length = wrong dimensions! The diagonal of a square relates to its side by $d = s\sqrt{2}$ , so you get incorrect face measurements. Always convert diagonal X to side length $s = \frac{X}{\sqrt{2}}$ first.

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

Why can't I just use X as the side of the square base?

Because X is the diagonal of the square base, not its side! The diagonal is always $\sqrt{2}$ times longer than the side. Using X directly would give you a completely wrong prism.

Which face diagonal is the question asking for?

It's asking for the diagonal of a rectangular face - the face that has one side equal to the square's side length and the other side equal to the prism's height (3X).

How do I remember the relationship between square side and diagonal?

Think of a square split by its diagonal into two right triangles. Using Pythagorean theorem: $s^2 + s^2 = d^2$ , so $d = s\sqrt{2}$ .

Why is the answer $X\sqrt{9.5}$ instead of a simpler form?

Because when you calculate $\sqrt{\frac{X^2}{2} + 9X^2} = X\sqrt{\frac{1}{2} + 9} = X\sqrt{9.5}$ , the 9.5 under the square root cannot be simplified further into a perfect square.

Can I solve this without using the Pythagorean theorem?

No! Since we need the diagonal of a rectangle, we must use the Pythagorean theorem. The diagonal is the hypotenuse of a right triangle formed by the rectangle's sides.

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Find the Face Diagonal of a Rectangular Prism with Base Diagonal X and Length 3X

3D Geometry with Square Base Diagonals

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just use X as the side of the square base?

Which face diagonal is the question asking for?

How do I remember the relationship between square side and diagonal?

Why is the answer $X\sqrt{9.5}$ instead of a simpler form?

Can I solve this without using the Pythagorean theorem?

🌟 Unlock Your Math Potential

More Questions

Using the Pythagorean Theorem in Cuboids

Calculate C¹B¹ in a Rectangular Prism with Diagonal √(6x²-12x+41)

Calculate Edge Length AE in a Rectangular Prism Using √(26a²+8a+16)

Calculate the Diagonal Length: Finding Distance in a Rectangular Prism with Sides 4b and 3a

Find the Space Diagonal Formula: Expressing Length in Terms of x, y, and z

Find the Face Diagonal of a Rectangular Prism with Base Diagonal X and Length 3X

3D Geometry with Square Base Diagonals

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just use X as the side of the square base?

Which face diagonal is the question asking for?

How do I remember the relationship between square side and diagonal?

Why is the answer X9.5 X\sqrt{9.5} X9.5​ instead of a simpler form?

Can I solve this without using the Pythagorean theorem?

🌟 Unlock Your Math Potential

More Questions

Using the Pythagorean Theorem in Cuboids

Calculate C¹B¹ in a Rectangular Prism with Diagonal √(6x²-12x+41)

Calculate Edge Length AE in a Rectangular Prism Using √(26a²+8a+16)

Calculate the Diagonal Length: Finding Distance in a Rectangular Prism with Sides 4b and 3a

Find the Space Diagonal Formula: Expressing Length in Terms of x, y, and z

Why is the answer $X\sqrt{9.5}$ instead of a simpler form?