Calculate Cube Height from 9 cm² Base Area: Geometric Problem

Q: Why is the height the same as the side length?

A cube is a special 3D shape where all edges are equal. Unlike a rectangular box, there's no difference between height, width, or depth - they're all the same measurement!

Q: How do I find the side length from the base area?

The base of a cube is a square, so base area = side × side. If base area = 9, then $ s^2 = 9 $, so $ s = \sqrt{9} = 3 $ cm.

Q: Could this be a rectangular box instead of a cube?

No! The problem specifically states it's a cube. If it were a rectangular box, we'd need more information since the height could be different from the base dimensions.

Q: How can I double-check my answer?

Square your answer to verify the base area: $ 3^2 = 9 \, \text{cm}^2 $ ✓. Also remember that in a cube, all faces have the same area!

Cube Dimensions with Given Base Area

Below is a cube with a base area equal to 9 cm².

Is it possible to calculate the height of the cube? If so, then what is it?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's calculate the height of the cube if we can.

00:12 In a cube, all sides are equal. We can call each side length, A.

00:17 First, let's work out the area of the cube's base.

00:21 Now, let's find the height of the cube, which is also A.

00:25 We'll take the square root to find the two possible values for A.

00:31 Remember, A has to be positive because it's the edge's length.

00:35 So, A is the height of the cube.

00:40 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Below is a cube with a base area equal to 9 cm².

Is it possible to calculate the height of the cube? If so, then what is it?

Step-by-step solution

To determine the height of the cube given the base area, follow these steps:

Step 1: Recognize that in a cube, all sides are equal, and thus the base area is given by the formula $s^2$ , where $s$ is the side length.
Step 2: We know the base area of the cube is $9 \, \text{cm}^2$ . Therefore, we have the equation $s^2 = 9$ .
Step 3: Solve for $s$ by taking the square root of both sides of the equation: $s = \sqrt{9}$ .
Step 4: Calculate $s = 3 \, \text{cm}$ .

Thus, the height of the cube is the same as the side length, which is $3 \, \text{cm}$ .

Therefore, the solution to the problem is $3 \, \text{cm}$ .

Final Answer

$3$

Key Points to Remember

Essential concepts to master this topic

Cube Property: All sides equal, so height equals side length
Technique: Find side from base area: $s = \sqrt{9} = 3$
Check: Verify base area: $3 \times 3 = 9 \, \text{cm}^2$ ✓

Common Mistakes

Avoid these frequent errors

Confusing cube height with other dimensions
Don't think the height is different from the side length = wrong answer! In cubes, height, width, and depth are identical. Always remember that all edges of a cube are equal.

Practice Quiz

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Find a,b

FAQ

Everything you need to know about this question

Why is the height the same as the side length?

A cube is a special 3D shape where all edges are equal. Unlike a rectangular box, there's no difference between height, width, or depth - they're all the same measurement!

How do I find the side length from the base area?

The base of a cube is a square, so base area = side × side. If base area = 9, then $s^2 = 9$ , so $s = \sqrt{9} = 3$ cm.

What if the base area isn't a perfect square?

You can still find the side length! For example, if base area = 8, then $s = \sqrt{8} = 2\sqrt{2}$ cm. The height would be $2\sqrt{2}$ cm too.

Could this be a rectangular box instead of a cube?

No! The problem specifically states it's a cube. If it were a rectangular box, we'd need more information since the height could be different from the base dimensions.

How can I double-check my answer?

Square your answer to verify the base area: $3^2 = 9 \, \text{cm}^2$ ✓. Also remember that in a cube, all faces have the same area!

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