Orthohedra Dimensions: Identifying the Cubic Case in 3D Geometry

Q: What exactly is an orthohedra?

An orthohedra is a 3D shape with rectangular faces and right angles at all corners. It's also called a rectangular prism or cuboid. A cube is a special type of orthohedra!

Q: Why can't 20, 7, 12 be a cube?

Because the three dimensions are different! For a cube, all sides must be equal. Since 20 ≠ 7 ≠ 12, this creates a rectangular box, not a cube.

Q: Do the dimensions have to be in any special order?

No! The order doesn't matter. Whether you write $12, 12, 12$ or rearrange them, it's still a cube because all three values are identical.

Q: What if two dimensions are the same but one is different?

That would make a rectangular prism, not a cube. For example, dimensions like $5, 5, 8$ create a shape that's square on two faces but rectangular on the others.

3D Shape Properties with Equal Dimensions

Which of the following dimensions of an orthohedra represents a cube?

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Step-by-step video solution

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00:00 Which box is a cube?

00:05 In a cube, all edges are equal

00:09 Let's check which option has equal edges

00:16 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

Which of the following dimensions of an orthohedra represents a cube?

Step-by-step solution

To determine which set of dimensions represents a cube, follow these steps:

Step 1: Review each set of dimensions.
Step 2: Check if all dimensions in a set are equal.

Now, let's evaluate each option:

Step 1: Analyze the given choices:
Choice 1: Dimensions $20, 7, 12$
Choice 2: Dimensions $7, 9, 15$
Choice 3: Dimensions $12, 12, 12$
Choice 4: Dimensions $8, 5, 6$

Step 2: Check for equality among dimensions in each choice:
- Choice 1: $20, 7, 12$ are all different. Not a cube.
- Choice 2: $7, 9, 15$ are all different. Not a cube.
- Choice 3: $12, 12, 12$ are all equal. This is a cube.
- Choice 4: $8, 5, 6$ are all different. Not a cube.

Therefore, the set of dimensions $12, 12, 12$ indicates a cube.

The correct answer is: $12,12,12$ .

Final Answer

$12,12,12$

Key Points to Remember

Essential concepts to master this topic

Cube Definition: All three dimensions (length, width, height) must be equal
Comparison Method: Check if all three numbers match: 12 = 12 = 12
Verification: Only one set has identical dimensions: $12, 12, 12$ ✓

Common Mistakes

Avoid these frequent errors

Confusing cubes with other rectangular solids
Don't assume any orthohedra with three dimensions is automatically a cube = wrong identification! A cube specifically requires ALL three dimensions to be identical. Always check that length = width = height before calling it a cube.

Practice Quiz

Test your knowledge with interactive questions

Identify the correct 2D pattern of the given cuboid:

FAQ

Everything you need to know about this question

What exactly is an orthohedra?

An orthohedra is a 3D shape with rectangular faces and right angles at all corners. It's also called a rectangular prism or cuboid. A cube is a special type of orthohedra!

Why can't 20, 7, 12 be a cube?

Because the three dimensions are different! For a cube, all sides must be equal. Since 20 ≠ 7 ≠ 12, this creates a rectangular box, not a cube.

Do the dimensions have to be in any special order?

No! The order doesn't matter. Whether you write $12, 12, 12$ or rearrange them, it's still a cube because all three values are identical.

What if two dimensions are the same but one is different?

That would make a rectangular prism, not a cube. For example, dimensions like $5, 5, 8$ create a shape that's square on two faces but rectangular on the others.

How do I remember what makes a cube special?

Think of dice! A standard die is a perfect cube because every face is a square with identical side lengths. All edges of a cube must be the same length.

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