Calculate the Diagonal Length AC in an 8cm × 4cm Cuboid

Q: Why can't I just add 8 + 4 to get the diagonal?

A diagonal is the shortest path across a shape, not around its edges! Adding gives you part of the perimeter. Use the Pythagorean theorem because the diagonal forms the hypotenuse of a right triangle.

Q: Should I simplify √80 or leave it as is?

Both forms are correct! $ \sqrt{80} $ is exact, while $ 4\sqrt{5} $ is simplified. Check what your teacher prefers, but don't convert to decimals unless specifically asked.

Q: What's the difference between face diagonal and space diagonal?

A face diagonal lies flat on one face of the cuboid (like AC here), while a space diagonal cuts through the interior from one corner to the opposite corner. This problem asks for a face diagonal.

3D Diagonal Calculation with Pythagorean Theorem

A cuboid has a width measuring 8 cm and a height of 4 cm.

Calculate the length of the side AC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate AC

00:03 The face of the box is a rectangle, therefore all angles are right angles

00:07 We'll use the Pythagorean theorem in triangle ACB to find AC

00:13 We'll substitute appropriate values according to the given data and solve for AC

00:31 Extract the root

00:37 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

A cuboid has a width measuring 8 cm and a height of 4 cm.

Calculate the length of the side AC.

Step-by-step solution

To find the length of the diagonal $AC$ on a cuboid, we will use the Pythagorean theorem twice:

Step 1: Calculate $AB$ , the diagonal of the face of the cuboid:

We assume the cuboid's known dimensions, with width $8 \, \text{cm}$ and height $4 \, \text{cm}$ . Assume $l$ for one dimension along the base.

Since $AB = \sqrt{w^2 + l^2}$ , solving for AB means understanding both directions. As the length isn't given, we solve specifically for vertically so $AC$ depends on full volume space:

Step 2: Calculate diagonal $AC$ :

$AC = \sqrt{AB^2 + \text{height}^2} = \sqrt{8^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80} \, \text{cm}$

In this way, we determine the length of diagonal $AC$ is $\sqrt{80}$ cm.

The correct choice corresponding to this calculation is Choice 3: $\sqrt{80}$ cm.

Final Answer

$\sqrt{80}$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Space diagonal uses $AC = \sqrt{width^2 + height^2}$
Technique: Calculate $AC = \sqrt{8^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80}$ cm
Check: Verify by squaring result: $(\sqrt{80})^2 = 80 = 64 + 16$ ✓

Common Mistakes

Avoid these frequent errors

Adding dimensions instead of using Pythagorean theorem
Don't simply add 8 + 4 = 12 cm! This gives the perimeter, not the diagonal length. Diagonals in rectangles and cuboids require the Pythagorean theorem. Always use $\sqrt{width^2 + height^2}$ for diagonal calculations.

Practice Quiz

Test your knowledge with interactive questions

A rectangular prism has a base measuring 5 units by 8 units.

The height of the prism is 12 units.

Calculate its volume.

FAQ

Everything you need to know about this question

Why can't I just add 8 + 4 to get the diagonal?

A diagonal is the shortest path across a shape, not around its edges! Adding gives you part of the perimeter. Use the Pythagorean theorem because the diagonal forms the hypotenuse of a right triangle.

Should I simplify √80 or leave it as is?

Both forms are correct! $\sqrt{80}$ is exact, while $4\sqrt{5}$ is simplified. Check what your teacher prefers, but don't convert to decimals unless specifically asked.

How do I know which dimensions to use for AC?

Look at the diagram carefully! Point A and C are connected by a diagonal that goes across the face of the cuboid. Use the width (8 cm) and height (4 cm) that form the rectangular face containing both points.

What's the difference between face diagonal and space diagonal?

A face diagonal lies flat on one face of the cuboid (like AC here), while a space diagonal cuts through the interior from one corner to the opposite corner. This problem asks for a face diagonal.

Why does my calculator give 8.94 for √80?

That's the decimal approximation! For exact answers in geometry, keep it as $\sqrt{80}$ . Decimal approximations are useful for real-world applications, but exact form is more precise for mathematical work.

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Calculate the Diagonal Length AC in an 8cm × 4cm Cuboid

3D Diagonal Calculation with Pythagorean Theorem

❤️ 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 add 8 + 4 to get the diagonal?

Should I simplify √80 or leave it as is?

How do I know which dimensions to use for AC?

What's the difference between face diagonal and space diagonal?

Why does my calculator give 8.94 for √80?

🌟 Unlock Your Math Potential

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