Calculate Volume Difference: Nested Cuboids with Dimensions 4×5×3

Q: What does 'the small cuboid fits 4 times in the large cuboid' actually mean?

It means you could place 4 copies of the small cuboid inside the large one without overlap. So if the small volume is 60 cm³, the large volume must be $ 4 \times 60 = 240 \text{ cm}^3 $.

Q: Why do we subtract the volumes instead of adding them?

The question asks for the remaining space - the volume of the large cuboid minus the small one inside it. Think of it as finding the empty space left over!

Q: How do I calculate the volume of a cuboid?

Use the formula: Volume = length × width × height. With dimensions 5, 4, and 3: $ V = 5 \times 4 \times 3 = 60 \text{ cm}^3 $

Q: What if the dimensions were given in different units?

Always convert to the same units first! If you mix cm and m, your volume units will be wrong. Keep everything in cm to get cm³, or everything in m to get m³.

Volume Relationships with Nested Cuboids

Given the small cuboid ABKD

inside the large cuboid

Given the small cuboid fits 4 times the large cuboid

BC=4 AB=5 BK=3

What is the volume of the large cuboid minus the small cuboid?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the volume difference between the boxes

00:03 We'll use the formula to calculate box volume

00:07 width times height times length

00:13 We'll substitute appropriate values and solve for the volume

00:29 This is the volume of the small box

00:34 The small box fits 4 times in the large box

00:38 We'll subtract the small box volume from the large box volume to find the difference

00:41 We'll use the volume value we found and solve for the difference

00:44 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

Given the small cuboid ABKD

inside the large cuboid

Given the small cuboid fits 4 times the large cuboid

BC=4 AB=5 BK=3

What is the volume of the large cuboid minus the small cuboid?

Step-by-step solution

To solve the problem, we'll start by calculating the volume of the small cuboid.

Step 1: Volume of the small cuboid
The small cuboid dimensions are $AB = 5$ , $BC = 4$ , and $BK = 3$ . Hence, its volume is:

$V = 5 \times 4 \times 3 = 60 \, \text{cm}^3$

Step 2: Volume of the large cuboid
Since the small cuboid fits four times into the large cuboid, the volume of the large cuboid is:

$4 \times 60 = 240 \, \text{cm}^3$

Step 3: Volume of the large cuboid minus the small cuboid
Subtracting the volume of the small cuboid from the large cuboid, we have:

$240 - 60 = 180 \, \text{cm}^3$

Thus, the volume of the large cuboid minus the small cuboid is $\boxed{180 \, \text{cm}^3}$ .

Final Answer

180 cm³

Key Points to Remember

Essential concepts to master this topic

Volume Formula: Multiply length × width × height for rectangular solids
Scaling Technique: When small fits 4 times in large: $4 \times 60 = 240$
Verification Check: Subtract volumes: $240 - 60 = 180 \text{ cm}^3$ ✓

Common Mistakes

Avoid these frequent errors

Confusing the scaling relationship direction
Don't think the large cuboid is 4 times smaller than the small one = getting 15 cm³! This reverses the relationship completely. Always remember: if the small fits 4 times in the large, then large volume = 4 × small volume.

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

What does 'the small cuboid fits 4 times in the large cuboid' actually mean?

It means you could place 4 copies of the small cuboid inside the large one without overlap. So if the small volume is 60 cm³, the large volume must be $4 \times 60 = 240 \text{ cm}^3$ .

Why do we subtract the volumes instead of adding them?

The question asks for the remaining space - the volume of the large cuboid minus the small one inside it. Think of it as finding the empty space left over!

How do I calculate the volume of a cuboid?

Use the formula: Volume = length × width × height. With dimensions 5, 4, and 3: $V = 5 \times 4 \times 3 = 60 \text{ cm}^3$

What if the dimensions were given in different units?

Always convert to the same units first! If you mix cm and m, your volume units will be wrong. Keep everything in cm to get cm³, or everything in m to get m³.

Can I solve this problem by finding the large cuboid's dimensions first?

That would be much harder! Since we don't know how the small cuboid fits inside (could be 4×1×1 arrangement, 2×2×1, etc.), it's easier to use the volume relationship directly.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Cuboids questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime