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

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

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?