Calculate Volume of Combined Cuboids: 3x4x7 Meter Orthohedra Assembly

Q: Why do I multiply the dimensions instead of adding them?

Volume measures 3D space! Adding gives you a linear measurement (like perimeter), but volume needs length × width × height to fill the entire box-shaped space.

Q: What if the cuboids overlap in the drawing?

The problem states we're using three separate orthohedra to build the body. Even if they appear connected, we calculate the total volume by adding all three individual volumes together.

Q: Do I need to worry about the arrangement shown in the diagram?

No! The total volume stays the same regardless of how the cuboids are arranged. Whether stacked, lined up, or configured differently, $ 3 \times 84 = 252 $ cubic meters.

Q: What does 'orthohedra' mean?

Orthohedra is just a fancy term for rectangular prisms or cuboids - boxes with six rectangular faces where opposite faces are identical and parallel.

Q: How do I check if 252 is reasonable?

Each cuboid is about the size of a small room ($ 84 \text{ m}^3 $). Three of them would be 252 cubic meters - roughly equivalent to a large house, which makes sense!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the volume of the combined body

00:04 We'll use the formula for calculating box volume

00:07 height multiplied by length multiplied by width

00:10 We'll substitute appropriate values and solve for the volume

00:13 This is the volume of one box, now we'll multiply this volume by the number of boxes

00:23 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

The dimensions of the cuboid are 3,4,7 meters

From three orthohedra of the same size we build the body in the drawing.

Calculates the volume of the created body

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Calculate the volume of a single cuboid using the dimensions provided.
Step 2: Multiply the volume of one cuboid by the number of cuboids to find the total volume of the assembled body.

Let's work through each step:

Step 1: The dimensions of the given cuboid are 3 meters, 4 meters, and 7 meters. The volume of this cuboid is calculated using the formula:

$\text{Volume of one cuboid} = \text{length} \times \text{width} \times \text{height} = 3 \times 4 \times 7$

Performing the calculation:

$3 \times 4 = 12$

$12 \times 7 = 84$

Therefore, the volume of one cuboid is $84 \text{ cubic meters}$ .

Step 2: Since there are three identical cuboids combined to form the body, we multiply the volume of one cuboid by 3:

$\text{Total Volume} = 84 \times 3$

Carrying out the multiplication:

$84 \times 3 = 252$

Therefore, the volume of the created body is $\text{252 cubic meters}$ .

Thus, the correct answer is $252$ , which matches choice 3 in the given options.

3

Final Answer

$252$

Key Points to Remember

Essential concepts to master this topic

Formula: Volume of cuboid equals length times width times height
Technique: Calculate single volume first: $3 \times 4 \times 7 = 84$
Check: Multiply single volume by quantity: $84 \times 3 = 252$ cubic meters ✓

Common Mistakes

Avoid these frequent errors

Adding dimensions instead of multiplying them
Don't calculate 3 + 4 + 7 = 14 for volume! This gives a linear measurement, not cubic space. Volume requires three-dimensional multiplication. Always multiply length × width × height for cuboid 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

Why do I multiply the dimensions instead of adding them?

+

Volume measures 3D space! Adding gives you a linear measurement (like perimeter), but volume needs length × width × height to fill the entire box-shaped space.

What if the cuboids overlap in the drawing?

+

The problem states we're using three separate orthohedra to build the body. Even if they appear connected, we calculate the total volume by adding all three individual volumes together.

Do I need to worry about the arrangement shown in the diagram?

+

No! The total volume stays the same regardless of how the cuboids are arranged. Whether stacked, lined up, or configured differently, $3 \times 84 = 252$ cubic meters.

What does 'orthohedra' mean?

+

Orthohedra is just a fancy term for rectangular prisms or cuboids - boxes with six rectangular faces where opposite faces are identical and parallel.

How do I check if 252 is reasonable?

+

Each cuboid is about the size of a small room ( $84 \text{ m}^3$ ). Three of them would be 252 cubic meters - roughly equivalent to a large house, which makes sense!

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Calculate Volume of Combined Cuboids: 3x4x7 Meter Orthohedra Assembly

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