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

00:00 Find the height of box BB1

00:03 Let's draw the diagonal of the base face

00:08 Use Pythagorean theorem in triangle A1B1C1 to find A1C1

00:11 Substitute appropriate values according to the given data and solve for the diagonal

00:19 This is the size of the diagonal in the square

00:28 Right angle between box diagonal and face

00:35 Now use Pythagorean theorem in triangle A1C1C to find CC1

00:42 Substitute appropriate values and solve for CC1

00:51 Isolate CC1

01:05 This is the height CC1

01:08 Each face in the box is a rectangle, therefore opposite sides are equal

01:12 And this is the solution to the question

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

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 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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