Calculate BB¹ in a Rectangular Prism with Diagonal Length 2√14 cm

Q: What exactly is a space diagonal in a rectangular prism?

A space diagonal connects two opposite corners of the prism, going through the interior. It's the longest possible straight line you can draw inside the prism, like from corner A to corner C¹.

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

The Pythagorean theorem works for two dimensions. For a 3D prism, you need the 3D extension: d² = l² + w² + h². Think of it as finding the diagonal of the base first, then using that with the height.

Q: How do I know which measurement is the height BB¹?

In the diagram, BB¹ is the vertical edge connecting point B (top face) to point B¹ (bottom face). It's perpendicular to both the length and width of the prism.

Q: What if I get a negative value under the square root?

This means there's an error in your setup. Check that the given diagonal is actually longer than both the length and width. A space diagonal must always be the longest measurement.

Q: Can I solve this problem in a different order?

Yes! You could find the face diagonal first using √(6² + 4²) = √52, then use d² = (face diagonal)² + h² to get 56 = 52 + h². Same answer!

Space Diagonal Formula with Height Calculation

Look at the rectangular prism in the figure below.

The length of the diagonal is $2\sqrt{14}$ cm.

Calculate $BB^1$ .

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's find the height of box B B 1.

00:14 First, draw the diagonal of the base face.

00:19 We'll use the Pythagorean theorem in triangle A 1 B 1 C 1 to find A 1 C 1.

00:25 Substitute the given values and solve for the diagonal.

00:30 This gives us the diagonal's size in the square.

00:39 Remember, there's a right angle between the box diagonal and the face.

00:46 Now, use the Pythagorean theorem in triangle A 1 C 1 C to find C C 1.

00:53 Substitute the values and solve for C C 1.

01:02 Isolate C C 1 to find its value.

01:16 This is the height, C C 1.

01:19 Since each face in the box is a rectangle, opposite sides are equal.

01:24 And that is how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the rectangular prism in the figure below.

The length of the diagonal is $2\sqrt{14}$ cm.

Calculate $BB^1$ .

Step-by-step solution

The problem asks us to find the height of a rectangular prism. Given the space diagonal $d = 2\sqrt{14}$ cm and base dimensions (length = 6 cm, width = 4 cm), we can use the formula for the space diagonal:

$d = \sqrt{l^2 + w^2 + h^2}$

Substitute the known values:

$2\sqrt{14} = \sqrt{6^2 + 4^2 + h^2}$

First, calculate $6^2 + 4^2$ :

$6^2 = 36, \quad 4^2 = 16$

Thus, $6^2 + 4^2 = 52$ .

Substitute back into the equation:

$(2\sqrt{14})^2 = 52 + h^2$

Calculate $(2\sqrt{14})^2$ :

$2^2 \times (\sqrt{14})^2 = 4 \times 14 = 56$

Substitute and solve for $h^2$ :

$56 = 52 + h^2$

Subtract 52 from both sides:

$h^2 = 4$

Take the square root of both sides:

$h = \sqrt{4} = 2$

Therefore, the height of the rectangular prism $BB^1$ is $2$ cm.

Final Answer

$2$

Key Points to Remember

Essential concepts to master this topic

Formula: Space diagonal = √(length² + width² + height²)
Technique: Square both sides: (2√14)² = 4 × 14 = 56
Check: Verify √(6² + 4² + 2²) = √56 = 2√14 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to square the coefficient when squaring expressions like 2√14
Don't calculate (2√14)² as 2 × 14 = 28! This ignores the coefficient squared and gives the wrong diagonal length. Always square both parts: (2√14)² = 2² × (√14)² = 4 × 14 = 56.

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 exactly is a space diagonal in a rectangular prism?

A space diagonal connects two opposite corners of the prism, going through the interior. It's the longest possible straight line you can draw inside the prism, like from corner A to corner C¹.

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

The Pythagorean theorem works for two dimensions. For a 3D prism, you need the 3D extension: d² = l² + w² + h². Think of it as finding the diagonal of the base first, then using that with the height.

How do I know which measurement is the height BB¹?

In the diagram, BB¹ is the vertical edge connecting point B (top face) to point B¹ (bottom face). It's perpendicular to both the length and width of the prism.

What if I get a negative value under the square root?

This means there's an error in your setup. Check that the given diagonal is actually longer than both the length and width. A space diagonal must always be the longest measurement.

Can I solve this problem in a different order?

Yes! You could find the face diagonal first using √(6² + 4²) = √52, then use d² = (face diagonal)² + h² to get 56 = 52 + h². Same answer!

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